after the third crash, i figured it wasn't a fluke, and tried again
with gdb. Just before section 8.8.7, there are a pair of multiline
equations. Go into the first, use the arrow keys to get out and in
from of the first (starts with der_f2), and enter it & move. *splat*
I've included the entire fille (207k), as i can't get it to happen in a
partial file (and had trouble repeating one more time in the full file).
(gdb) run
Starting program: /usr/src/lyx-1_0_x/src/lyx.mm
In MapColor [flcolor.c 816] ColormapFull. Using subsititutions
LyX: Couldn't get color linen
Program received signal SIGSEGV, Segmentation fault.
0x8112a44 in MathedXIter::ipop (this=0x822db44) at math_iter.C:803
803 crow = crow->next;
(gsb) bt
#0 0x8112a44 in MathedXIter::ipop (this=0x822db44) at math_iter.C:803
#1 0x8122952 in MathedCursor::SetPos (this=0x8255da8, x=-4, y=116)
at math_cursor.C:313
#2 0x80e786b in InsetFormula::Edit (this=0x834f0e8, x=0, y=0) at formula.C:483
#3 0x80b5c1a in LyXFunc::Dispatch (this=0x8255218, ac=39,
do_not_use_this_arg=0x0) at lyxfunc.C:2268
#4 0x80ac19c in LyXFunc::processKeyEvent (this=0x8255218, ev=0x4007f2ec)
at lyxfunc.C:325
#5 0x8154eb5 in LyXView::KeyPressMask_raw_callback (fl=0x824d2a0,
xev=0x4007f2ec) at LyXView.C:337
#6 0x400383bf in fl_register_raw_callback ()
#7 0x40037758 in fl_last_event ()
#8 0x40037ee2 in fl_treat_interaction_events ()
#9 0x40037f1c in fl_check_forms ()
#10 0x8057a57 in LyXGUI::runTime (this=0x82365e8) at lyx_gui.C:629
#11 0x804dfe5 in LyX::LyX (this=0xbffffbfc, argc=0xbffffca0, argv=0xbffffcb4)
at ../src/lyx_main.C:128
#12 0x804d88c in main (argc=1, argv=0xbffffcb4) at ../src/main.C:51
(gdb) q
The program is running. Exit anyway? (y or n) n
Not confirmed.
(gdb) c
Continuing.
lyx: SIGSEGV signal caught
Sorry, you have found a bug in LyX. If possible, please read 'Known bugs'
under the Help menu and then send us a full bug report. Thanks!
lyx: Attempting to save document /home/hawk/Dissertation/dissertation.lyx as...
1) /home/hawk/Dissertation/dissertation.lyx.emergency
Save seems successful. Phew.
Bye.
Program received signal SIGABRT, Aborted.
0x401d1601 in kill ()
(gdb) c
Continuing.
Program terminated with signal SIGABRT, Aborted.
The program no longer exists.
(gdb)
#This file was created by <hawk> Thu Apr 1 16:44:20 1999
#LyX 1.0 (C) 1995-1999 Matthias Ettrich and the LyX Team
\lyxformat 2.15
\textclass report
\begin_preamble
\usepackage{isuthesis}
\usepackage{verbatim}
\usepackage[normalem]{ulem}
\newcommand{\rh}{\uwave}
\newcommand{\so}{\sout}
\newcommand{\jd}{\uline}
\newcommand{\wk}{\uuline}%
\newcommand{\jim}{\textsf}
%\renewcommand{\jim}{}
%\renewcommand{\rh}{}
%\renewcommand{\jd}{}
%\renewcommand{\wk}{}
%\renewcommand{\so}{}
\end_preamble
\language default
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\paperfontsize default
\spacing single
\papersize letterpaper
\paperpackage a4
\use_geometry 1
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\paperorientation portrait
\leftmargin 1in
\topmargin 1in
\rightmargin 2.25in
\bottommargin 0.1in
\secnumdepth 2
\tocdepth 2
\paragraph_separation indent
\defskip medskip
\quotes_language english
\quotes_times 2
\papercolumns 1
\papersides 1
\paperpagestyle default
\layout Title
Numerical optimization of recursive systems of equations
\layout Author
Richard E.
Hawkins
\layout Abstract
Investigation of numerical methods for the optimization of recursive systems,
as found in breeding with partial genetic information.
\layout Abstract
\latex latex
\backslash
jim{Changes from Jim's suggestions look like this}
\layout Abstract
\latex latex
\backslash
wk{Changes from Wolfgang's suggestions look like this}
\layout Abstract
\latex latex
\backslash
jd{Changes from Jack's suggestions look like this}
\layout Abstract
\latex latex
\backslash
so{Removed text looks like this}
\layout Abstract
\latex latex
\backslash
rh{Rick's changes look like this}
\layout Abstract
Investigation of numerical methods for the optimization of recursive systems,
as found in breeding with partial genetic information.
\layout Standard
\begin_inset LatexCommand \tableofcontents{}
\end_inset
\layout Chapter
Introduction
\layout Standard
\latex latex
\backslash
pagenumbering{arabic}
\layout Standard
Recent years have seen rapid progress in computational technology, genetics,
and the animal breeding industry, among others.
While computer speed and storgage have increased,
\latex latex
\backslash
jd{
\latex default
advances in molecular genetics have made it
\latex latex
}
\latex default
possible to test individual animals for the presence of specific genes,
and
\latex latex
\backslash
so{
\latex default
breeders have
\latex latex
}
\backslash
jd{
\latex default
The breeding industry has
\latex latex
}
\latex default
become more concentrated.
These advances can be combined to find more efficient methods of improving
genetic progress within breeding herds.
\begin_float margin
\layout Standard
jack: I don't know.
where?
\end_float
Particularly, refinements of the long-established method of dynamic programming
to search disjoint subspaces allow the use of genetic testing for individual
genes to maximize genetic progress, even when such maximization is analytically
impossible.
\layout Standard
Within living memory,swine were born, raised, lived, and slaughtered primarily
on the family farm.
They were mostly bred with the farmer's own herd, or with an impressive
hog from another nearby farmer.
Slaughter would take place on the farm, where the assorted parts of the
carcass would be preserved for personal consumption.
\bar under
\latex latex
\layout Standard
At the turn of the century, the majority of hogs were sent by rail to a
handful of regional slaughter and packing houses to feed the growing population
of the cities
\begin_inset LatexCommand \cite[ p. 4-5]{Vertical Coordination}
\end_inset
.
Rather than feeding the family, raising hogs for market was a way to begin
or expand a farm with relatively small amounts of capital.
Still, though, a herd could be maintained and improved by avoiding obvious
inbreeding and breeding the sows to the
\latex latex
\backslash
sout{
\latex default
easily recognizable superior
\latex latex
}
\latex default
sires
\latex latex
\backslash
jd{
\latex default
with superior phenotypic characteristics
\latex latex
}
\latex default
.
\begin_float margin
\layout Standard
add: --
\begin_inset Quotes eld
\end_inset
the outwordly observable traits.
\begin_inset Quotes erd
\end_inset
?
\end_float
\layout Standard
The world has changed since then, and hog production with it.
Increasingly, additional traits of the hog were seen to have a genetic
basis.
Some were to be avoided by careful breeding, while others were to be sought
out.
Research has found better ways to breed the hog, and the predominant family
operation of farrow to finish is being replaced by a system with separate
operations for farrowing, nurseries, and finishing
\begin_inset LatexCommand \cite[ p. 4]{Vertical Coordination}
\end_inset
.
For some, it became profitable to buy sucklings to raise for market.
\layout Standard
While the raising of swine remained primarily a family operation, the production
of breeding stock rapidly became a concentrated industry.
\latex latex
\backslash
so{
\latex default
Brands arose for these commercial breeders, and t
\latex latex
}
\latex default
Today the market is dominated by breeding companies,
\latex latex
\backslash
jd{
\latex default
each aggressively marketing their own brand of suckling
\latex latex
}
\latex default
.
\bar under
\bar default
These companies are constantly positioning for market share, and can either
command a better price, or take a larger share, or both, by a relatively
modest improvement.While a tenth of a per cent of additional meat per hog
may not have been noticeable to a farmer early in the century, commercial
farmers see an effect similar to Rockefeller's reducing the number of tacks
used in barrels: a fraction of a penny per hog can add up.
Rockefeller saved $60,000 per year by reducing the number of tacks per
barrel by one, and a commercial operation reaps a measurable amount by
the slightest improvement--and losses for the slightest defect.
\layout Standard
The market has changed further since then.
While artificial insemination of swine was not practical even ten years
ago, today **% of hogs are conceived this way
\begin_float margin
\layout Standard
Jawn Lawrence survey
\end_float
\begin_inset LatexCommand \cite{Christian}
\end_inset
.
While sales in the past were based merely on animal weight, today price
is adjusted for carcass quality.
Statistical sampling and proxy measures such as the thickness of fat on
the back of a hog create price premiums, whereas a excess fat will bring
a penalty.
\begin_float margin
\layout Standard
cite?
\end_float
In a multi-stage operation it is now necessary for each manager to be able
to assess the quality of both inputs and outputs, creating further competitive
pressure for the industries.
\layout Standard
While genetics have been indirectly recognized as affecting
\latex latex
\backslash
jd{
\latex default
the performance of
\latex latex
}
\latex default
animals
\latex latex
\backslash
so{
\latex default
since
\latex latex
}
\latex default
long before Mendel's age, resulting in selective breeding, the development
of modern genetics and molecular biology has meant that an increasing number
of genes that affect quality in various manners have been
\latex latex
\backslash
so{
\latex default
found.
\latex latex
}
\latex default
\latex latex
\backslash
jd{
\latex default
identified.
\latex latex
}
\latex default
The estrogen receptor gene
\begin_float margin
\layout Standard
xref to example?
\end_float
is known to increase litter size
\begin_inset LatexCommand \cite{Rothschild}
\end_inset
, while a stress gene causing fainting
\latex latex
\backslash
jd{
\latex default
and reduced meat quality
\latex latex
}
\latex default
has also been identified.Not only are these genes recognized, but an individual
animal may be tested
\emph on
prior
\emph default
to breeding to avoid less desirable progeny.
\layout Standard
While the famous example of Mendel's Peas concerned a
\emph on
qualitative trait,
\emph default
\begin_inset LatexCommand \index{qualitative trait}
\end_inset
or one that is either present or not present, most genes of interest
\latex latex
\backslash
jd{
\latex default
in livestock genetics
\latex latex
}
\latex default
concern
\emph on
quantitative traits
\emph default
\begin_inset LatexCommand \index{qualitative trait}
\end_inset
, or those which take a range of values, such as animal size.
Rather than fully governing outcome, a gene may be one of thousands which
\latex latex
\backslash
jd{
\latex default
, along with the environment,
\latex latex
}
\latex default
contributes to the size of an animal, some with relatively minor individual
effect, and others with strong effect.
If these more important genes can be recognized, the possibility exists
of identifying a better rule for selecting animals to breed, yielding an
increase in quality without an accompanying increase in cost.
\layout Standard
Better selection decisions have a direct economic impact.
An improvement in the current generation also improves all subsequent generatio
ns--an extra 1% profit from improved genetics in this year's production
raises all future years.
which in turn increases the value of the farm by 1%.
That is, the change is permanent--as
\latex latex
\backslash
so{
\latex default
well as
\latex latex
}
\backslash
jd{
\latex default
is the increase in
\latex latex
}
\latex default
the value of the farm
\latex latex
\backslash
so{
\latex default
, by 1%
\latex latex
}
\latex default
.
More importantly, with the improved genetics, the wealth of the operation
is improved by a quantifiable amount.
\layout Standard
Current breeding methods are based on
\emph on
phenotypic
\emph default
\begin_inset LatexCommand \index{phenotype}
\end_inset
information, the observable traits of an animal.
From this information, a
\emph on
breeding value
\emph default
\begin_inset LatexCommand \index{breeding value}
\end_inset
is estimated, or
\latex latex
\backslash
jd{
\latex default
the
\latex latex
\latex default
statistical
\latex latex
}
\latex default
expected value of the
\latex latex
\backslash
jd{
\latex default
collective
\latex latex
}
\latex default
effect of the genes passed by the animal to its progeny.
The goal is
\latex latex
\backslash
jd{
\latex default
to maximize rates of
\latex latex
}
\latex default
genetic improvement, an approach which is closely related methodologically
to that of optimizing profits; in fact, it is a subset.
In the simplest case, revenue is strictly a multiple of the quantity produced--
pounds of milk, for example--and the cost of testing is very low.
In these cases, revenue is a simple multiple of the breeding value, production
costs are taken as constant, and the optima for genetic improvement are
also the economic optima.
\layout Standard
More complicated cases can be handled as well.
Quality as well as quantity can be handled: leaner pork may fetch a higher
price per hundredweight.
Sales volume may depend upon quality: there is more demand, as well as
a higher price, for superior semen for artificial insemination.
Finally, there may be a
\begin_inset Quotes eld
\end_inset
brand
\begin_inset Quotes erd
\end_inset
premium in having fixed a gene as present in the breeder's animals that
exceeds the direct value of the gene.
As with revenues, costs may not be constant.
\begin_float margin
\layout Standard
Jim: not just in this example
\end_float
It will frequently be the case that testing for the presence of genes in
individual animals imposes a significant cost, and that the number of generatio
ns to test becomes a variable to optimize.
The quality of the animal may increase or lower costs, as well: a sow with
larger litters may have increased veterinary costs, partially offsetting
the gains, while optimizing for disease resistance could be expected to
reduce the costs of raising the breeding herd and commercial animals.
\layout Section
The Problem
\layout Standard
While Mendel created a discipline
\begin_float margin
\layout Standard
Jim questions
\begin_inset Quotes eld
\end_inset
discipline
\begin_inset Quotes erd
\end_inset
\end_float
with wrinkled and unwrinkled peas, today's geneticist faces
\latex latex
\backslash
so{
\latex default
harder
\latex latex
}
\latex default
\latex latex
\backslash
jim{
\latex default
more
\latex latex
complex}
\latex default
problems.
Creatures have many traits, most of which are influenced by large numbers
of genes--enough that they may frequently be treated as having infinite
count.
In recent years an increasing number of these genes have been identified.
\latex latex
\backslash
so{
\latex default
but usually
\latex latex
}
\backslash
jim{
\latex default
However, they still
\latex latex
}
\latex default
work in concert with a large number of as yet undiscovered genes
\begin_inset LatexCommand \cite{need cite}
\end_inset
.
\layout Standard
Given that a gene contributing significantly to a quantitative trait of
interest can be detected, it seems likely that this information can be
used to improve the herd.
Particularly, the best possible improvement
\emph on
using
\emph default
the information will be
\emph on
at least
\emph default
as good as without the information.
The question is then
\emph on
how
\emph default
to use the genetic information.
The question as to
\emph on
how
\emph default
to use the information is not easy; the first proposed rules, using
\begin_inset Quotes eld
\end_inset
genotypic selection,
\begin_float margin
\layout Standard
More explanation?
\end_float
\begin_inset Quotes erd
\end_inset
have found
\emph on
lower
\emph default
long-term performance using the genetic information
\begin_inset LatexCommand \cite{Gibson}
\end_inset
.
\layout Standard
If a gene is designated as
\begin_inset Formula \( B \)
\end_inset
\latex latex
\backslash
jim{
\latex default
when
\latex latex
}
\latex default
present, and
\begin_inset Formula \( b \)
\end_inset
\latex latex
\backslash
jim{
\latex default
when
\latex latex
}
\latex default
not present, there are three
\latex latex
\backslash
jim{
\latex default
\begin_inset Quotes eld
\end_inset
genotypes,
\begin_inset Quotes erd
\end_inset
or
\latex latex
}
\begin_float margin
\layout Standard
quotes or
\emph on
italics
\emph default
for terms?
\end_float
types of animals:
\begin_inset Formula \( bb \)
\end_inset
,
\begin_inset Formula \( bB \)
\end_inset
, and
\begin_inset Formula \( BB \)
\end_inset
.
A naive approach would be to breed only the
\begin_inset Formula \( BB \)
\end_inset
's, assuming that the gene is desirable.
However, this is far from optimal.
Suppose that each
\begin_inset Formula \( B \)
\end_inset
is worth
\begin_inset Formula \( 1 \)
\end_inset
, and that the unknown genes yield a standard-normal distribution for the
trait.
Approximately 5% of the
\begin_inset Formula \( bb \)
\end_inset
's will draw a value greater than
\begin_inset Formula \( 2 \)
\end_inset
from the distribution, while half of the
\begin_inset Formula \( BB \)
\end_inset
's will draw a negative number.
That 5% is clearly more desirable than the lower half; it is desirable
to keep some of each.
\layout Standard
The question remains, however, as to the optimal combination.
By selectively breeding, and with a litter size of 10, a gene can be brought
from a frequency of 5% to 99% in about 5 generations
\begin_inset LatexCommand \ref{Results}
\end_inset
.
This means that if the program were to last for ten generations, with only
the last generation of concern, the first five could be spent merely looking
for high values from the normal distribution, with the last five spent
increasing the frequency of the
\begin_inset Formula \( BB \)
\end_inset
gene.
This is not the optimal pattern, but is offered to show that the animals
without the favored gene remain of value in maximizing the trait in question.
Further, a choice must be made as to which
\begin_inset Formula \( BB \)
\end_inset
's to breed.
\layout Standard
While many classes of breeding rules exist, those to be considered here
will select animals by
\emph on
truncation selection
\begin_inset LatexCommand \index{truncation selection}
\end_inset
\emph default
within each of the
\emph on
genotypes
\begin_inset LatexCommand \index{genotype}
\end_inset
\emph default
, such as
\begin_inset Formula \( bB \)
\end_inset
: All creatures above the threshold quality, or
\emph on
estimated breeding value
\emph default
\begin_inset LatexCommand \index{breeding value}
\end_inset
\begin_inset LatexCommand \index{estimated breeding value}
\end_inset
, within that genotype breed, and are mated randomly amongst all breeding
creatures.
\begin_float footnote
\layout Standard
Faster progress could be made by selecting mates.
However, this would introduce concerns about inbreeding, making the problem
far more complicated.
It is prudent to first solve the simple problem, and then approach the
more complicated problem with the knowledge gained.
\end_float
Further, it will usually be assumed that the herd is arbitrarily large.
As such, the mean breeding value
\begin_inset LatexCommand \index{breeding value}
\end_inset
in the following generation, as well as all other variables of concern,
are degenerate random variables.
\layout Standard
Preliminary work has already been done in this area.
Dekkers, and van Arendonk, consider the case of a single
\latex latex
\backslash
jim{
\emph on
\latex default
quantitative trait locus,
\emph default
or
\latex latex
}
\latex default
QTL,
\latex latex
\backslash
jim{
\latex default
at which a detectable gene is located,
\latex latex
}
\latex default
and use optimal control to find the maximum improvement in the final generation
of an infinite herd
\begin_inset LatexCommand \cite{Dekkers}
\end_inset
.
\layout Standard
Two basic scenarios, a small finite number of periods, and an infinite horizon
using consider the case of a single QTL, and use optimal control to find
the maximum improvement in the final generation of an infinite herd
\begin_inset LatexCommand \cite{Dekkers}
\end_inset
.
\layout Standard
Two basic scenarios, a small finite number of periods, and an infinite horizon
using present (discounted) value
\begin_inset LatexCommand \index{present value}
\end_inset
, will be used, and each will be examined both with optimal control and
numeric methods.
In the simplest model, only the genetic improvement over time is considered.
This problem is computationally harder, due to stiff Hessian matrices,
and is of value in testing the robustness of the algorithms.
Furthermore, an analytic solution is known from the work of Dekkers and
van Arendonk
\begin_inset LatexCommand \cite{Dekkers}
\end_inset
.
\layout Subsection
The Breeding Industry
\layout Standard
\latex latex
\backslash
jim{
\latex default
add something about this
\latex latex
}
\layout Subsection
Economic Model
\layout Standard
\latex latex
\backslash
jim{
\latex default
The breeding model translates directly to an
\latex latex
}
\latex default
economic model by
\latex latex
\backslash
so{
\latex default
can be made by
\latex latex
}
\latex default
using the present value for a finite number of generations.
Revenues are the present discounted value of some function, possibly linear,
of the trait in question plus any brand premium,
\begin_inset Formula
\begin{equation}
\sum ^{\infty }_{t=0}\rho ^{t}\left[ R(\bar{A}_{t})+B\left( \bar{A}_{t},p_{t}\right)
\right]
\end{equation}
\end_inset
\begin_float margin
\layout Standard
make
\begin_inset Formula \( \bar{A} \)
\end_inset
a function?
\end_float
, where the revenue
\begin_inset Formula \( R \)
\end_inset
reflects the value of animals of the grade indicated by the breeding value,
\begin_inset Formula \( \bar{A}_{t} \)
\end_inset
\latex latex
\backslash
jim{
\latex default
in generation
\begin_inset Formula \( t \)
\end_inset
which depends upon gene frequency
\begin_inset Formula \( p_{t} \)
\end_inset
and other cators
\latex latex
}
\latex default
, and the brand premium
\begin_inset Formula \( B \)
\end_inset
is an amount, possibly zero, paid beyond the revenue at that same level
when the gene is fixed, i.e., gene frequency
\begin_inset Formula \( p_{t}=1 \)
\end_inset
.
\layout Standard
Costs may be divided into three significant pieces: fixed costs, costs based
on the quality of the herd, and testing costs.
As it is assumed that the size of the herd is already chosen, and that
a full herd will be kept, fixed costs may be ignored in all cases
\latex latex
\backslash
jim{
\latex default
, as they are the same
\latex latex
}
\latex default
.
Cost is discounted in the same manner as revenue.
\layout Standard
\latex latex
\backslash
so{
\latex default
Quality cost is a marginal cost, measuring from the fixed cost,that may
be either positive or negative.
If the objective of breeding is to increase litter size, this cost will
likely be greater than zero, due to increased veterinary costs for the
extra offspring.
However, if the object is disease resistance, veterinary costs should drop
as the breed becomes stronger.
\latex latex
}
\layout Standard
\latex latex
\backslash
jim{
\latex default
Quality cost is a marginal cost, measuring from the fixed cost,that may
be either positive or negative.
Disease resistance, for example will reduce vetrinary costs, yielding a
negative quantity cost.
On the other hand, increasing litter size will increase costs, as there
are more animcals to feed and house.
This increased cost, however, should be more than offset by the increased
revenue from the extra animals.
Similarly, breeding for increased milk yield or animal size may cause the
animals to eat more, raising costs while producing aditional salable product.
\latex latex
}
\layout Standard
The final type of cost is that of testing.
\latex latex
\backslash
so{
\latex default
In some cases, testing may be continued forever, and treated as a fixed
cost.
However, in many real-world cases, the cost is significant.
\latex latex
\latex default
It is likely that testing may be valuable for several generations, after
which the gene is to be fixed in the population.
This should happen once future gains from testing exceed future costs.
\latex latex
}
\latex default
\latex latex
\backslash
jim{
\latex default
If testing were to continue forever, it could be treated as a fixed cost
of the enterprise.
However, this is unlikely: unless testing can be done by a casual visual
inspection, such as a differnt color of animal, it will have at least
\emph on
some
\emph default
cost.
Once the gains from use of this genetic information, as compared to the
program which would be used without the information, exceed the cost of
testing, testing will cease.
\latex latex
}
\layout Standard
Combining these
\latex latex
\backslash
jim{
\latex default
revenue streams and costs
\latex latex
}
\latex default
yields the value of the enterprise,
\begin_inset Formula
\begin{equation}
\sum ^{\infty }_{t=0}\rho ^{t}\left[ R(\bar{A}_{t})+B\left( \bar{A}_{t},p_{t}\right)
-F_{t}-Q\left( \bar{A}_{t},p_{t}\right) -T\left( t\right) \right]
\end{equation}
\end_inset
which is to be maximized
\latex latex
\backslash
jim{
\latex default
, where
\begin_inset Formula \( \rho \)
\end_inset
is the discount factor,
\begin_inset Formula \( F_{t} \)
\end_inset
is the fixed cost in that time period,
\begin_inset Formula \( Q \)
\end_inset
the quality cost, and
\begin_inset Formula \( T \)
\end_inset
the cost of testing.
Only models in which
\begin_inset Formula \( T \)
\end_inset
does not change until testing is halted, after which it becomes
\begin_inset Formula \( 0 \)
\end_inset
, will be considered; strategies such as testing alternate generations will
be ignored.
\latex latex
}
\layout Subsection
Analytic Solutions
\layout Standard
\begin_float margin
\layout Standard
Jim: i don't know.
Delete section?
\end_float
In the simplest case, with a finite horizon, Dekkers and van Arendonk
\begin_inset LatexCommand \cite{Dekkers}
\end_inset
have used optimal control theory to show an analytic solution, solving
recursively from the final period.
Chapter [
\begin_inset LatexCommand \ref{sec:analyticSolutions}
\end_inset
] finds that an analytic solution does not exist for even the simplest case
with an infinite horizon--while the solution exists, it has one too many
variables to solve.
\layout Subsection
Computational Solutions
\layout Standard
In the case of a finite number of generations, simple second order methods
are inadequate to solve the problem (
\latex latex
\backslash
S
\latex default
\begin_inset LatexCommand \ref{NR/SFG}
\end_inset
,
\emph on
infra)
\emph default
.
\begin_float margin
\layout Standard
Jim: ??
\end_float
Choices made in the early generations have very little weight in the final
generation, and the Hessian matrix becomes stiff.
\begin_inset LatexCommand \index{matrices, stiff}
\end_inset
This can be solved by augmented Newton methods
\begin_inset LatexCommand \index{Newton Methods, Augmented}
\end_inset
, a genetic algorithm
\begin_inset LatexCommand \index{genetic algorithm}
\end_inset
, or both.
\layout Standard
The infinite horizon presents a simpler problem in some ways.
The discounting of the future reduces the stiffness found in the simple
model--but will make the
\begin_inset Quotes eld
\end_inset
far off
\begin_inset Quotes erd
\end_inset
generations nearly irrelevant, rather than early generations.
The key is in determining how many generations are necessary to be a
\begin_inset Quotes eld
\end_inset
large number.
\begin_inset Quotes erd
\end_inset
That is, is it fifteen, a hundred, or five hundred generations that must
be considered to approximate infinity?
\layout Standard
Two broad categories of computational solutions are sought: conclusions
to partial analytic solutions, and
\begin_inset Quotes eld
\end_inset
brute force
\begin_inset Quotes erd
\end_inset
methods which can handle methods not at all amenable to analytic solution.
The first category is preferable when possible, but will probably require
analytic work for each variation of the problem proposed.
However, these methods will use known and established methods to complete
problems, and will be certain of achieving optimal solutions.
\latex latex
\backslash
so{
\latex default
However the
\latex latex
}
\latex default
\latex latex
\backslash
jim{
\latex default
Unfortuneately
\latex latex
}
\latex default
, such partial solutions generally do not exist for problems of economic
interest.
Accordingly, finding methods for the second class are also desirable.
In some cases, it may be possible to provide a proof of convergence for
a class of problems, and in others it may not.
However, the methods likely have value even when proof is impossible: while
it may not be possible to guarantee optimality, it remains simple to compare
the proposed solution to the best
\emph on
existing
\emph default
solution--for a
\latex latex
\backslash
so{
\latex default
real
\latex latex
}
\latex default
breeder, an improvement upon current output is valuable, even if there
is an unknowable better improvement.
\layout Section
Objectives
\layout Standard
\latex latex
\backslash
so{
\latex default
A series of results are
\latex latex
}
\latex default
\latex latex
\backslash
jim{
\latex default
The results
\latex latex
}
\latex default
expected
\latex latex
\latex default
from this line of research
\latex latex
\backslash
jim{
\latex default
are
\latex latex
}
\latex default
:
\layout Enumerate
An evaluation of the economic value of genetic information
\layout Enumerate
The devlopment of an approach to quantiatively evaluate gentic improvements
in a herd.
\layout Enumerate
Application of the program to one or more specific cases of identified pork
genes.
\layout Standard
To achieve these objectives, the following sub-objectives must be accomplished:
\layout Enumerate
A concise formulation of the recursive optimization problem.
\layout Enumerate
An algorithm or methodology which can find the optimal values for the control
variables.
\layout Enumerate
An implementation of this algorithm for the specific case of maximizing
the present discounted value of a herd in which one or more QTL's have
been identified, considering price premiums for quality and the cost of
testing.
The program code for this algorithm should be as modular as possible to
allow re-use for other problems.
\layout Enumerate
\latex latex
\backslash
so{
\latex default
Application of the program to one or more specific cases of identified pork
genes.
\latex latex
}
\layout Chapter
\latex latex
\backslash
so{
\latex default
Literature Review
\latex latex
}
\layout Section
\latex latex
\backslash
so{
\latex default
Genetics
\latex latex
}
\layout Section
\latex latex
\backslash
so{
\latex default
Pork Industry
\latex latex
}
\layout Section
\latex latex
\backslash
so{
\latex default
Analytic Method
\latex latex
}
\layout Standard
\begin_float margin
\layout Standard
should this even be here?
\end_float
\layout Section
\latex latex
\backslash
so{
\latex default
Computational Methods
\latex latex
}
\layout Standard
\latex latex
\backslash
so{
\latex default
The literature is nearly empty of applicable models, essentially stopping
with linear-quadratic problems.
\latex latex
}
\layout Subsubsection
\latex latex
\backslash
so{
\latex default
Linear-Quadratic
\begin_inset LatexCommand \index{linear-quadratic}
\end_inset
\latex latex
}
\layout Standard
\latex latex
\backslash
so{
\latex default
refholding spot
\latex latex
}
\layout Subsubsection
\latex latex
\backslash
so{
\latex default
Quasi-Newton
\begin_inset LatexCommand \index{quasi-Newton}
\end_inset
\latex latex
}
\layout Subsubsection
\latex latex
\backslash
so{
\latex default
Genetic Algorithms
\begin_inset LatexCommand \index{Genetic Algorithm}
\end_inset
\latex latex
}
\layout Chapter
Genetics
\layout Standard
Some standard assumptions are made while working with theoretical genetics.
These assumptions will generally apply to very large groups as well.
Fundamentally, they are large sample results from the Central Limit Theorem,
at such a sample size that variance of the population mean.has dropped to
zero.
\layout Section
Structure of the Model
\layout Standard
Given an initial population in period
\begin_inset Formula \( 0 \)
\end_inset
, it is sought to maximize the average phenotypic value
\begin_inset Formula \( B \)
\end_inset
of the animals
\begin_inset Formula \( T \)
\end_inset
generations later, where the phenotypic value is the total effect from
genotype
\begin_inset Formula \( g \)
\end_inset
, polygenic value
\begin_inset Formula \( A \)
\end_inset
, and environment
\begin_inset Formula \( E \)
\end_inset
:
\begin_inset Formula
\begin{equation}
B=g+A+E
\end{equation}
\end_inset
\begin_inset Formula \( E \)
\end_inset
will be assumed to be
\begin_inset Formula \( 0 \)
\end_inset
in all cases.
\layout Section
The Breeding Value and Total Genetic Value
\layout Standard
There are two contributors to the breeding value: the major genes, and the
polygenes.
As the population is arbitrarily large, each range of values is present
in its statistically expected value.
It is assumed that it is the mean of the phenotypic value for the entire
population that is of interest, rather than the traits of individuals.
In a large population, the average breeding value and phenotypic value
are the same.
For example, it is the milk production of a
\latex latex
\backslash
jim{
\latex default
dairy
\latex latex
}
\latex default
herd that is most relevant, rather than how much a particular cow produces.
\begin_float footnote
\layout Standard
Taken largely from
\begin_inset LatexCommand \cite{Dekkers}
\end_inset
, pp **.
\end_float
The discussion that follows considers only a single locus with two alleles,
for the sake of simplicity.
Except as noted, the concepts carry over directly to the case of multiple
loci.
\layout Standard
These genotypes are tagged by the variable
\begin_inset Formula \( m \)
\end_inset
, taking the values 0, 1, and 2, referring to the number of times the favorable
allele is present.
\begin_inset Formula \( g_{m} \)
\end_inset
refers to the genotypic value of the gene, and takes the values
\begin_inset Formula \( \left\{ -a,0,a\right\} \)
\end_inset
for
\begin_inset Formula \( m=\{0,1,2\} \)
\end_inset
\latex latex
\backslash
jim{
\latex default
in the case of additive major genes, which will be the primary case considered.
An asymmetric responce, such as
\begin_inset Formula \( \left\{ -b,0,a\right\} \)
\end_inset
is also possible.
\latex latex
}
\begin_float margin
\layout Standard
switch order of b,a?
\end_float
\layout Standard
\latex latex
\backslash
jim{
\latex default
Allowing
\begin_inset Formula \( i \)
\end_inset
to index individual animals,
\latex latex
}
\latex default
the polygenic breeding values of
\latex latex
\backslash
jim{
\latex default
an
\latex latex
}
\latex default
animal can be estimated by
\begin_inset Formula \( \hat{A}_{imt} \)
\end_inset
, the heritability of the trait not accounting for the major gene.
In the absence of a stochastic contribution, this estimate is the actual
value.
\begin_inset Formula \( \hat{A}_{imt} \)
\end_inset
is represented as a deviation from
\begin_inset Formula \( \bar{A}_{t} \)
\end_inset
.
\begin_inset Formula
\[
\hat{A}_{imt}=h^{2}\left( P-g\right) \]
\end_inset
\latex latex
\backslash
jim{
\latex default
where
\begin_inset Formula \( P \)
\end_inset
is the observed phenotypic value of the animal, and
\begin_inset Formula \( h^{2} \)
\end_inset
is the
\emph on
heritability
\emph default
\begin_inset LatexCommand \index{heritability}
\end_inset
, or the portion of polygenic value which is passed to the next generation
\latex latex
.}
\emph on
\latex default
\emph default
Letting
\begin_inset Formula \( b_{mt} \)
\end_inset
be a weight for genotype
\begin_inset Formula \( m \)
\end_inset
in generation
\begin_inset Formula \( t \)
\end_inset
, the animal has a selection value
\begin_inset Formula
\begin{equation}
\label{simpleselectionvalue}
I_{imt}=b_{mt}g_{m}+\hat{A}_{imt}
\end{equation}
\end_inset
For genotypic selection,
\begin_inset Formula \( b_{mt}=1 \)
\end_inset
.
In the absence of gametic phase disequilibrium
\begin_inset LatexCommand \index{disequilibrium, gametic phase}
\end_inset
, or correllation between polygenic value and the major gene, the variance
of the total genetic value is the same for each of the three groups, while
the mean is different, resulting in distributions as seen in Figure
\begin_inset LatexCommand \ref{simpleselectionvalue}
\end_inset
\layout Standard
\begin_float fig
\layout Standard
\align center
\begin_inset Figure size 208 174
file graphics/threedists.ps
width 4 70
angle -90
flags 9
\end_inset
\layout Caption
Three Distributions With Same Variance
\begin_inset LatexCommand \label{threedists}
\end_inset
\end_float
\layout Standard
In each generation, the frequency of the major gene is expressed by
\begin_inset Formula \( p_{t} \)
\end_inset
.
A value of 0 would mean that the entire population were homozygotes without
the favorable allele, and a value of 1 would mean that the entire population
had it twice.
In equilibrium, there will be
\begin_inset Formula \( p_{t}^{2} \)
\end_inset
homozygotes with the favorable allele twice
\begin_inset Formula \( (AA) \)
\end_inset
,
\begin_inset Formula \( 2p_{t}(1-p_{t}) \)
\end_inset
heterozygotes
\begin_inset Formula \( (Aa) \)
\end_inset
, and
\begin_inset Formula \( (1-p_{t})^{2} \)
\end_inset
homozygotes without it
\begin_inset Formula \( (aa) \)
\end_inset
.
Expressing the average polygenic breeding value of the population as
\begin_inset Formula \( \bar{A}_{t} \)
\end_inset
, the average breeding value can be expressed as
\begin_inset Formula
\begin{equation}
\bar{G}_{t}=a(2p_{t}-1)+\bar{A}_{t}
\end{equation}
\end_inset
Also,
\begin_inset Formula \( \bar{A}_{mt} \)
\end_inset
refers to the average polygenic value of animals with major genotype
\begin_inset Formula \( m \)
\end_inset
in generation
\begin_inset Formula \( t \)
\end_inset
.
\layout Standard
Combining, the state of the system at time
\begin_inset Formula \( t \)
\end_inset
is fully described by the values
\begin_inset Formula \( \{\bar{A}_{t},p_{t},\sigma ^{2}_{pt},\mu _{pt}\} \)
\end_inset
where
\begin_inset Formula \( \mu _{pt} \)
\end_inset
and
\begin_inset Formula \( \sigma ^{2}_{pt} \)
\end_inset
are the mean and variance of the polygenic distribution at time
\begin_inset Formula \( t \)
\end_inset
.
If disequilibrium is considered, then this set must be expanded to account
for the fact that the polygenic distribution is different for each of the
three genotypes, and the set becomes
\begin_inset Formula \( \{\overline{A}_{t},p_{t},\sigma _{mpt},\mu _{mpt}\} \)
\end_inset
.
Finally, if optimization is being done with a finite time horizon, the
number of remaining generations becomes important, and
\begin_inset Formula \( T-t \)
\end_inset
is needed as well.
\layout Standard
The problem is then to choose values for
\begin_inset Formula \( b_{mt} \)
\end_inset
to maximize
\begin_inset Formula \( A_{T} \)
\end_inset
, which is homomorphic with choosing
\begin_inset Formula \( x_{mt} \)
\end_inset
or
\begin_inset Formula \( f_{mt} \)
\end_inset
.
\layout Section
Standard Types of Breeding
\layout Subsubsection
Truncation
\layout Standard
Selection is by truncation of the breeding intensity
\begin_inset Formula \( I \)
\end_inset
.
For each
\begin_inset Formula \( m \)
\end_inset
, a cutoff point
\begin_inset Formula \( x_{mt} \)
\end_inset
is chosen.
Animals with
\begin_inset Formula \( I_{imt}>x_{mt} \)
\end_inset
are selected to breed for the next generation, while the rest do not breed.
The breeding animals then randomly choose mates.
\begin_float footnote
\layout Standard
Choosing mates for them increases problems with inbreeding of other genes.
\end_float
This cutoff point can also be expressed as a fraction,
\begin_inset Formula \( f_{mt} \)
\end_inset
, which describes the portion of animals of that type bred.
Then
\begin_inset Formula
\begin{equation}
f_{mt}=1-F_{mt}(x_{mt})
\end{equation}
\end_inset
where
\begin_inset Formula \( F_{mt} \)
\end_inset
is the cumulative distribution function for
\begin_inset Formula \( I_{fmt} \)
\end_inset
.
\layout Subsubsection
Mass Selection
\layout Standard
In mass selection,
\begin_inset Formula \( b_{mt}=h^{2} \)
\end_inset
for all
\begin_inset Formula \( m \)
\end_inset
.
Thus
\begin_inset Formula
\begin{equation}
I=h^{2}g_{m}+h^{2}\left( P-g_{m}\right) =h^{2}P
\end{equation}
\end_inset
The genotypic information is simply ignored, and the same truncation point
is used for each genotype..
This is the simplest selection method.
\layout Subsubsection
Genotypic Selection
\layout Standard
For genotypic selection,
\begin_inset Formula \( b_{mt}=1 \)
\end_inset
for all
\begin_inset Formula \( m \)
\end_inset
.
This is an initial attempt to take the genotypes into account.
Thus
\begin_inset Formula
\begin{equation}
I=1g_{m}+h^{2}\left( P-g_{m}\right) =\left( 1-h^{2}\right) g_{m}+h^{2}P
\end{equation}
\end_inset
\layout Subsection
disequilibrium
\layout Standard
After a selection with different cutoff points for the different genotypes,
a negative correlation between the polygenic values and the major genotypes
of parents will exist.
This disequilibrium is not currently addressed, and needs to be added.
\layout Subsection
Variance
\layout Standard
Simple models assume that changes in the mean breeding value are small,
and that therefore changes in the variance of the breeding orders are of
second order smallness, and need not be considered.
However, the models considered here attempt to maximize the change that
can be made, and this would appear to no longer be a reasonable assumption.
The changes in the mean are a function of the variance.
If the variance drops significantly due to selection, gains from selection
will be overstated
\latex latex
\backslash
jim{
\latex default
, as polygenic improvement is proportonal to polygenic variance.
\latex latex
}
\layout Standard
There are two sources of change to the variance of the polygenic distribution.
The first is the Bulmer effect
\begin_inset LatexCommand \index{Bulmer effect}
\end_inset
\begin_inset LatexCommand \cite[pp. 126-131]{Bulmer}
\end_inset
, which is a reduction in variance due to gametic phase disequilibrium.
Breaking this down,
\begin_float margin
\layout Standard
I expect we strike or totally rewrite this.
Only discuss changeif we do 5-d model.
\end_float
\begin_inset Formula
\begin{equation}
\sigma _{gt}^{2}=p_{t}\left( 1-p_{t}\right)
\end{equation}
\end_inset
\begin_inset Formula
\begin{eqnarray}
\sigma ^{2}_{pt} & = & E\left[ P^{2}_{t}-E\left[ P_{t}\right] ^{2}\right] \\
E\left[ P_{t}\right] & = & p_{t}^{2}E\left[ P_{t}^{2}|m=2\right] +2p_{t}\left(
1-p_{t}\right) E\left[ P_{t}^{2}|m=1\right] +\left( 1-p_{t}\right) ^{2}E\left[
P_{t}^{2}|m=0\right]
\end{eqnarray}
\end_inset
where
\begin_inset Formula \( E\left[ \right] \)
\end_inset
is the expectation operator.
The total variance can be expressed as
\begin_inset Formula
\begin{equation}
\sigma ^{2}_{At}=\sigma ^{2}_{gt}+\sigma ^{2}_{pt}+2\rho _{t}\sigma _{gt}\sigma _{pt}
\end{equation}
\end_inset
where the subscripts
\begin_inset Formula \( A \)
\end_inset
,
\begin_inset Formula \( g \)
\end_inset
, and
\begin_inset Formula \( p \)
\end_inset
indicated the breeding, genotypic, and polygenic distributions.
While this can be taken further, it suffices for the present to observe
that the variance is changing due to selection, and that the simple model
does not account for this.
\layout Chapter
The Pork Industry
\layout Chapter
Dynamic Programming
\layout Standard
\begin_float margin
\layout Standard
absorb portions of Methods into this chapter
\end_float
Computational Methods
\layout Standard
The literature is nearly empty of applicable models, essentially stopping
with linear-quadratic problems.
\layout Subsubsection
Linear-Quadratic
\begin_inset LatexCommand \index{linear-quadratic}
\end_inset
\layout Standard
refholding spot
\begin_inset LatexCommand \label{scantlit}
\end_inset
\layout Subsubsection
Quasi-Newton
\begin_inset LatexCommand \index{quasi-Newton}
\end_inset
\layout Subsubsection
Genetic Algorithms
\begin_inset LatexCommand \index{Genetic Algorithm}
\end_inset
\layout Chapter
\latex latex
\backslash
jim{
\latex default
The Model
\latex latex
}
\layout Section
\latex latex
\backslash
jim{
\latex default
The Analytic Problem
\latex latex
}
\layout Standard
Mathematically, these problems
\latex latex
\backslash
so{
\latex default
of interest
\latex latex
}
\latex default
present uncommon problems for both analytic and numerical optimization.
Particularly, the recursive
\begin_inset LatexCommand \index{recursive}
\end_inset
nature of the choice space twists the meanings and relevance of the commonly
used jacobian and Hessian matrices, as well as making the choice space
more difficult to define.
In fact, there is almost no literature for numerical methods unless the
entire system can be well approximated as linear-quadratic (see
\begin_inset LatexCommand \ref{scantlit}
\end_inset
).
\begin_float margin
\layout Standard
fix this ref--to dp literature
\end_float
\layout Section
General Description of
\latex latex
\backslash
so{
\latex default
Covered
\latex latex
}
\latex default
Models
\layout Standard
A single class of models will be considered, although the definition of
this class is large enough to cover topics of interest in several fields.
Within this class of model, application will be made to genetic selection
rules.
\layout Enumerate
A known initial state,
\begin_inset Formula \( S_{0} \)
\end_inset
, fully describing the system at the initial time,
\begin_inset Formula \( t_{0} \)
\end_inset
.
\layout Enumerate
A choice set,
\begin_inset Formula \( \mathcal{A}_{t} \)
\end_inset
, for all time periods, which can be expressed as a function of
\begin_inset Formula \( S_{t} \)
\end_inset
.
\layout Enumerate
Determinism.
Given the prior state and choice made, it must be possible to calculate
the current state.
\begin_float footnote
\layout Standard
There is no reason that the state cannot include informations from prior
state
\end_float
\layout Enumerate
A single-valued fitness function for the entire model.
This function should be at least piecewise continuous.
\begin_float margin
\layout Standard
where should discussion of this go?
\end_float
\layout Subsection*
The State
\layout Standard
The system must have a fully defined state for each time period.
This state includes all of the variables for that time period, and if relevant,
those from prior time periods.
Among these variables are those used to determine the state of the following
generation, and one or more which are used for that period's contribution
to the total fitness.
Note that for a model with a finite number of generations, the state will
generally include the number of generations remaining, and thus such models
always lack the Markov property
\begin_inset LatexCommand \index{Markov property}
\end_inset
.
\layout Subsection*
The Choice Space
\layout Standard
In each model, there is a choice set in each time period, which itself is
determined by choices made in prior periods.
That is, if
\begin_inset Formula \( \mathcal{C}_{t} \)
\end_inset
is the set of choices, or action set
\begin_inset LatexCommand \index{action set}
\end_inset
, available at time
\begin_inset Formula \( t \)
\end_inset
, and
\begin_inset Formula \( c_{t} \)
\end_inset
is the choice actually made,
\begin_inset Formula \( \mathcal{C}_{t} \)
\end_inset
itself is determined by
\begin_inset Formula \( \left\{ c_{s}:s<t\right\} \)
\end_inset
,
\latex latex
\backslash
jim{
\latex default
as in Figure [
\begin_inset LatexCommand \ref{choicespacefig1}
\end_inset
]
\latex latex
}
\latex default
.
That is,
\layout Standard
\begin_inset Formula
\begin{equation}
\mathcal{C}_{t}=\mathcal{C}_{t}\left( c_{t-1}|\mathcal{C}_{t-1}\left( c_{t-2}\ldots
c_{0}|\mathcal{C}_{0}\right) \right)
\end{equation}
\end_inset
\begin_float fig
\layout Standard
\begin_inset Figure size 129 132
file choices.eps
flags 1
\end_inset
\layout Caption
\begin_inset LatexCommand \label{choicespacefig1}
\end_inset
The choice taken determines the next choice space
\end_float
\layout Standard
This
\latex latex
\backslash
so{
\latex default
gives the difficulty
\latex latex
that}
\latex default
\latex latex
\backslash
jim{
\latex default
complicates matters, as
\latex latex
}
\latex default
a change at time
\begin_inset Formula \( s \)
\end_inset
means that there is not only an interaction between
\begin_inset Formula \( c_{t} \)
\end_inset
and
\begin_inset Formula \( c_{s} \)
\end_inset
, which can be handled by considering cross partial derivatives, but that
\begin_inset Formula \( c_{s} \)
\end_inset
actually
\emph on
determines
\emph default
\begin_inset Formula \( \mathcal{C}_{t} \)
\end_inset
, and thus to speak coherently about effects from changing
\begin_inset Formula \( c_{s} \)
\end_inset
require calculating the changes in
\begin_inset Formula \( \left\{ c_{t},\mathcal{C}_{t}:t>s\right\} \)
\end_inset
that result.
\layout Standard
\begin_float margin
\layout Standard
move these 2 paragraphs?
\end_float
The
\emph on
state
\emph default
of the system can be fully described by the number of periods remaining
in the program (if finite), the current frequency of the gene identified,
or
\emph on
major gene
\emph default
\begin_inset LatexCommand \index{major gene}
\end_inset
, the mean of the unknown genes, or
\emph on
polygenes
\emph default
\begin_inset LatexCommand \index{polygene}
\end_inset
, the polygenic variance, and the correlation between the genotype and polygenic
distribution.
\layout Standard
Each period, a choice is made as to the proportion of each group to be bred
must be made.
An additional choice of whether or not to test can be made, as well: it
may well be the case that at some level of improvement, the benefits of
the information no longer outweigh the cost.
\layout Subsection*
Determinism
\layout Standard
While a stochastic model would certainly have interest, it is first necessary
to find appropriate tools for a deterministic system.
\layout Subsection*
Objective Function
\layout Standard
It is necessary to have a single objective function for the entire model
to optimize.
This function could be a value for a single period, or it can be some weighted
combination.
While continuity is desirable, it is not strictly necessary.
However, it must be finite valued at all locations.
Genetic algorithms generally have, within limits, some ability to move
across discontinuities and to escape from local optima.
\begin_float margin
\layout Standard
need ref
\end_float
\layout Section
Formulation
\layout Standard
At this time, a simplified version of the genetic model will be used, including
some strong assumptions.
Particularly, an infinitely large herd with an infinite number of genes
is assumed, yielding a normal distribution of the breeding values of the
animals.
It is further assumed that genetic progress does not cause deviations from
normality.
Initially, the variance of the distribution will be assumed to be fixed,
though this restriction will eventually be relaxed.
\begin_float margin
\layout Standard
will it?
\end_float
\layout Standard
For the case of optimization of genetic improvement with a QTL, the general
model can be further specified.
The time periods are the generations of breeding, and the state variables
are the frequency of the major genes (including covariance), mean and variance
of the polygenic distribution, the covariance of the polygenic and major
genes (if included), and the breeding value (which is actually a function
of the others).
The major genes and polygenic effect will be assumed to interact only in
a linear manner.
\begin_float margin
\layout Standard
include a discussion of this?
\end_float
\layout Subsection*
Simple Case: One Locus
\layout Standard
In each generation, there will be the same number of
\begin_inset Quotes eld
\end_inset
kinds
\begin_inset Quotes erd
\end_inset
of creature, as determined by the combinations of major genes.
With single locus with only two possible values, there are three types
of creatures, namely those with 0, 1, or 2 of the gene.
Generally, the number of types is the product of the number of permutations
for each locus.
\layout Standard
The choice set for each generation is the fraction of each kind of creature
to be bred to produce the following generation.
The sum of the products of these fractions with the frequency of that type
of creature must equal
\latex latex
\backslash
jim{
\latex default
\begin_inset Formula \( Q \)
\end_inset
\latex latex
}
\latex default
the fraction of the entire population needed to produce the next generation:
\begin_inset Formula
\begin{equation}
\label{genericqcons}
\sum _{m}p_{mt}f_{mt}=Q
\end{equation}
\end_inset
Note that these choices are not fully independent; if there are
\begin_inset Formula \( n \)
\end_inset
types, the first
\begin_inset Formula \( n-1 \)
\end_inset
choices also determine the final choice.
Additionally, while the fractions are necessarily in the
\begin_inset Formula \( \left[ 0,1\right] \)
\end_inset
range, not all choices are necessarily possible.
For example, if
\begin_inset Formula \( .2 \)
\end_inset
of the population is needed to breed the following generation, and type
\begin_inset Formula \( m \)
\end_inset
has a frequency of
\begin_inset Formula \( .4 \)
\end_inset
,
\begin_inset Formula \( f_{mt} \)
\end_inset
must be chosen from
\begin_inset Formula \( \left[ 0,.5\right] \)
\end_inset
.
\layout Standard
Given the multivariate distribution of the major and polygenes,
\begin_float margin
\layout Standard
\begin_inset Quotes eld
\end_inset
major and polygenes
\begin_inset Quotes erd
\end_inset
or
\begin_inset Quotes eld
\end_inset
major and poly- genes
\begin_inset Quotes erd
\end_inset
or ??
\end_float
and the frequencies of the kinds, the average polygenic breeding value
may be calculated.
\begin_float margin
\layout Standard
include formula here?
\end_float
Alternatively, and more easily, it can be calculated from it's prior value.
For example, with a single major gene, and ignoring gametic phase disequilibriu
m
\begin_inset LatexCommand \index{disequilibrium, gametic phase}
\end_inset
, the result is
\begin_inset Formula
\begin{equation}
\bar{A}_{t+1}=\bar{A}_{t}+\frac{\sigma _{t}}{Q}\sum q_{mt}z_{mt}
\end{equation}
\end_inset
\latex latex
\backslash
jim{
\latex default
where
\begin_inset Formula \( z_{mt} \)
\end_inset
is the height of the standard normal distribution at the truncation point.
\latex latex
}
\begin_float margin
\layout Standard
is
\begin_inset Formula \( \sigma _{t} \)
\end_inset
correct?
\begin_inset Formula \( \sigma _{mt} \)
\end_inset
?
\begin_inset Formula \( h^{2} \)
\end_inset
\begin_inset Formula \( \sigma \)
\end_inset
\end_float
Similarly, the total breeding value
\begin_inset LatexCommand \index{breeding value, total}
\end_inset
,
\begin_inset Formula \( \bar{G}_{t} \)
\end_inset
can be calculated from
\begin_inset Formula \( \bar{A}_{t} \)
\end_inset
and the gene frequencies at time
\begin_inset Formula \( t \)
\end_inset
.
\layout Standard
Finally, the objective function will typically be one of two forms: a function,
perhaps equality, of
\begin_inset Formula \( \bar{G}_{T} \)
\end_inset
, the value after the final breeding, or a sum of a discounted profit function
of the values of the state variables in all generations.
The first is appropriate when measuring the maximum genetic progress in
a given number of generations.
The latter calculates the economic value of a breeding program.
Note that it is not necessarily the breeding value alone which is used,
but more likely a function of the breeding value.
One such case could be a price premium for animals which exceed a certain
quality.
In this case, the breeding value might represent the amount of meat produced,
which could be sold at different prices depending upon how lean it is.
Another example would be a fixed price premium for the fixation of a gene
in the population, which would introduce a discontinuity into the objective
function.
\layout Subsection*
Simplified Finite Number of Generations
\layout Standard
In this simplified version, only the mean breeding value of the herd after
the final breeding generation is considered.
This is a test of the maximum rate of progress over a finite period, but
is not economically reasonable: it neglects both the sale of the animals
during most of the program, and the residual value of the operation.
Further, the present value of the early generations should count for
\emph on
more
\emph default
than the final generation, rather than nothing.
\layout Standard
Nonetheless, this model is useful in developing the numerical methods, and
ferrets out potential problems in the methods.
It also shows the maximum rate at which genetic progress
\emph on
could
\emph default
be made in a fixed number of generations, which is almost surely not the
same solution as the economic problem.
Additionally, this is the problem for which analytical solution has been
demonstrated, and is therefore useful as a check on the accuracy of the
methods.
\begin_inset LatexCommand \cite{Dekkers}
\end_inset
\begin_float margin
\layout Standard
should the solution be included?
\end_float
\layout Subsection*
Discounted Finite Generations
\layout Standard
The objective function is then the net present value of all future profits,
discounted for all periods considered in the model.
\layout Standard
It should be noted that the discounted finite generations problem is not
in itself of interest; it is developed as a tool for the infinite horizon
problem.
\begin_float margin
\layout Standard
cross reference this?
\end_float
\layout Standard
The simplest form is to simply discount the revenues of each generation,
assuming that costs are fixed and that revenues are a linear function of
revenue, e.g., that the breeding value represents milk produced.
In this case, the problem is to maximize total profit
\begin_inset Formula
\begin{equation}
\pi =\sum _{t=0}^{T}\left( 1-r\right) ^{t}\bar{G}_{t}
\end{equation}
\end_inset
\latex latex
\backslash
jim{
\latex default
where
\begin_inset Formula \( r \)
\end_inset
\latex latex
\latex default
is the discount rate, and is equal to
\begin_inset Formula \( 1-\rho \)
\end_inset
\latex latex
}
\latex default
Note that the first generation can be left out of the summation, as profits
during that generation are predetermined by the initial state.
However, as more complicated cases may have variable costs from choices
made, this generation will be left in for the sake of consistency.
\layout Standard
This formulation, however, still only accounts for a very simple case of
a single major gene affecting a single trait.
Further, it does not take into account economic factor such as premiums
for fixed gene lines, profits, or the ability to terminate testing as a
choice variable.
To account for such abilities, profitability should be considered.
For example, consider a simple case in which a hog has both meat yield
\begin_inset Formula \( Y \)
\end_inset
and leanness
\begin_inset Formula \( Z \)
\end_inset
.
At a fixed leanness, or quality,
\begin_inset Formula \( Z \)
\end_inset
, revenues are presumably linear in
\begin_inset Formula \( Y \)
\end_inset
.
However, there is no
\emph on
a priori
\emph default
reason to believe that revenue is linear in leanness, though it would presumabl
y be increasing within some range of interest, and then likely decreasing.
Assuming that the desirability of leanness is unimodal, or that there is
a single most desirable value, with desirability decreasing above and below
this value, weak quasiconcavity should apply to revenues as a function
of
\begin_inset Formula \( Z \)
\end_inset
.
Finally, it is possible that for a given set of genes for yield, that total
meat produced may be different for different levels of leanness.
Revenue then becomes a function of leanness, and the problem expands to
be that of maximizing
\begin_inset Formula
\begin{equation}
\pi =\sum ^{T}_{t=0}\left( 1-t\right) ^{t}\pi _{t}\left( Y_{t},Z_{t}\right)
\end{equation}
\end_inset
Note that this formulation includes the possibility that costs for change
with the size or leanness of the animal.
Letting
\begin_inset Formula \( \gamma \)
\end_inset
denote the vector of total genetic and breeding values, this becomes
\begin_inset Formula
\begin{equation}
\pi =\sum ^{T}_{t=0}\left( 1-t\right) ^{t}\pi \left( \gamma _{t}\right)
\end{equation}
\end_inset
\layout Standard
Finally, some models will include a price premium or penalty based on the
presence of a gene.
To remain general, let
\begin_inset Formula \( \theta \)
\end_inset
denote the vector of all gene frequencies and their covariances, and the
problem becomes
\begin_inset Formula
\begin{equation}
\label{finsumstate}
\pi =\sum ^{T}_{t=0}\left( 1-t\right) ^{t}\pi \left( \gamma _{t},\theta _{t}\right)
\end{equation}
\end_inset
\begin_inset LatexCommand \label{finprofofstat}
\end_inset
\layout Standard
While these optimization problems express the profit function
\begin_inset Formula \( \pi \)
\end_inset
as a function of the state variables, the determinism of the model means
that the state variables themselves are functions of the choice variables,
\begin_inset Formula \( f_{mt} \)
\end_inset
.
Letting
\begin_inset Formula \( \phi \)
\end_inset
be the vector of fractions selected, equation
\begin_inset LatexCommand \ref{finsumstate}
\end_inset
becomes,
\begin_inset Formula
\begin{equation}
\label{finsumphi}
\pi =\sum ^{T}_{t=0}\left( 1-t\right) ^{t}\pi \left( \phi _{t}\right)
\end{equation}
\end_inset
\layout Standard
The final modification is to note that the cessation of testing for one
or more genes may be included in the model.
That is, it is entirely possible that the a point may be reached at which
the value of the information from continued testing is less than the cost.
Consideration will be limited to cases in which testing occurs in every
generation until terminated.
As such, the testing choice variable for a gene takes a whole number value.
Letting
\begin_inset Formula \( \tau _{m} \)
\end_inset
indicate the the first generation without testing for gene
\begin_float margin
\layout Standard
don't use m.
It indicates type.
What variable to index loci with?
\end_float
\begin_inset Formula \( m \)
\end_inset
, and
\begin_inset Formula \( \tau \)
\end_inset
itself be the entire vector,
\begin_inset LatexCommand \ref{finsumphi}
\end_inset
becomes
\begin_inset Formula
\begin{equation}
\label{finsumtest}
\pi =\sum ^{T}_{t=0}\left( 1-t\right) ^{t}\pi \left( \phi _{t},\tau \right)
\end{equation}
\end_inset
It should be noted that selecting
\begin_inset Formula \( f_{mt} \)
\end_inset
after testing stops is nonsensical; there is only one type after this point,
from which all must be chosen.
\layout Subsection*
Infinite Horizon
\layout Standard
This is the actual economic problem of interest.
As an economic model, the herd should be assumed to continue forever--even
though the farmer will eventually retire, the discounted value of the remaining
infinite horizon reflects the value for which he can sell the herd.
This problem is actually easier to solve analytically than the finite horizon--
in the cases in which it
\emph on
is
\emph default
soluble.
However, unless the problem can be analytically reduced to a single equation
or function, it is not possible to solve for an infinite number of generations;
a rule must be found for approximation of this horizon.
\layout Standard
The problem is not as futile as it sounds.
With the introduction of discounting, far off
\begin_float margin
\layout Standard
\begin_inset Quotes eld
\end_inset
far off
\begin_inset Quotes erd
\end_inset
? colloquial
\end_float
generations have an increasingly diminished impact.
Thus it can be expected that a convergence
\begin_inset LatexCommand \index{convergence}
\end_inset
theorem can be written to the effect that for any desired
\begin_inset Formula \( \epsilon \)
\end_inset
, and for any generation
\begin_inset Formula \( T \)
\end_inset
, that
\begin_inset Formula \( N_{\epsilon ,T} \)
\end_inset
can be chosen sufficiently large that
\begin_inset Formula
\begin{equation}
\left| f_{mt}^{N_{\epsilon ,T}}-f_{mt}\right| <\epsilon \forall t\leq T
\end{equation}
\end_inset
\layout Standard
\begin_float footnote
\layout Standard
\latex latex
\backslash
so{
\latex default
old paragraph: Even in the non-discounted finite case, solutions for adjacent
numbers of generations strongly resemble each other.
The early generations for ten generations are quite similar to those for
fifteen
\begin_inset LatexCommand \ref{Results}
\end_inset
,which will already be quite close to infinite.
\latex latex
}
\end_float
Furthermore, the introduction of a generation in which to cease testing
as a choice variable simplifies, rather than complicates, the problem.
Once testing ends, selection is by mass selection, in which the animals
to breed are selected solely on the observable value of the trait.
The genetic progress under mass selection is well known, and thus once
the gene is fixed (or even not fixed, but testing ended), there is nothing
new to the problem.
A
\begin_inset Quotes eld
\end_inset
canned
\begin_inset Quotes erd
\end_inset
function can be written for the value of the entire future after the cutoff
generation, as discussed at []
\begin_float margin
\layout Standard
ref
\end_float
.
This changes the problem from
\begin_inset Quotes eld
\end_inset
find choices for all generations forever,
\begin_inset Quotes erd
\end_inset
with an infinite number of choice variables, to
\begin_inset Quotes eld
\end_inset
choose a finite number of generations, and choices for those generations.
\begin_inset Quotes erd
\end_inset
\layout Standard
Even without such a cutoff, the problem remains tractable.
Gene frequencies That is, after a very small number of generations, the
gene frequency becomes very close to one, and remains so permanently, which
closely approximates mass selection, given that essentially all of the
herd have become homozygotes with the gene.
Accordingly, an infinite horizon may be simulated by considering
\begin_inset Quotes eld
\end_inset
enough
\begin_inset Quotes erd
\end_inset
generations
\layout Section
Genetic Models
\layout Standard
Very little consideration will be given to purely genetic models without
economic consequence.
In fact, there is only one possible genetic question that can be answered,
namely
\begin_inset Quotes eld
\end_inset
What is the greatest genetic progress possible in
\begin_inset Formula \( N \)
\end_inset
generations?
\begin_inset Quotes erd
\end_inset
where
\begin_inset Formula \( N \)
\end_inset
is a fixed number.
Any other question must involve a state from at least two generations after
the initial state, which means that these must be weighted.
Barring Divine Revelation, the question of how to weight generations is
inherently a question of relative economic value.
\layout Section
Analytic Methods for Soluble Cases
\layout Standard
\begin_inset LatexCommand \label{sec:analyticSolutions}
\end_inset
There are categories of cases for which it is possible to partially solve
analytically, such as the case considered in
\begin_inset LatexCommand \cite{Dekkers}
\end_inset
.
However, even in this case, analytic methods fail to yield a complete solution,
but instead take the problem to a point at which iterative methods can
solve for the truncation points.
This is as far as such a method can get.
\layout Standard
However, Dekkers et al solved for a finite case.
The actual economic problem is for the infinite horizon, in which the business
continues indefinitely.
The problem can easily be reformulated to include discounting and an infinite
horizon.
Consider again the Dekkers model.
The genetic value
\begin_inset Formula \( G_{t} \)
\end_inset
is defined as the sum of the effects of the major gene and the polygenic
value
\begin_inset Formula
\begin{equation}
\label{siminfobjfn}
\bar{G}_{t}=a\left( 2p_{t}-1\right) +\bar{A}_{t}
\end{equation}
\end_inset
where
\begin_inset Formula \( p_{t} \)
\end_inset
is the frequency of the major gene
\begin_inset LatexCommand \index{gene frequency}
\end_inset
, or the portion of loci in the population that actually have this gene,
and
\begin_inset Formula \( a \)
\end_inset
is the of
\emph on
each copy
\emph default
of the gene.
\begin_inset Formula \( \bar{A}_{t} \)
\end_inset
\emph on
\emph default
is the mean of the polygenic value, which is assumed to be normally distributed
with standard deviation
\begin_inset Formula \( \sigma \)
\end_inset
\emph on
.
\emph default
The available choice variables are
\begin_inset Formula \( \left\{ f_{mt}:m\in \{0,1,t\},t\in \{0,1,...T-1\right\} \)
\end_inset
, the fraction of type
\begin_inset Formula \( m \)
\end_inset
that will be bred in generation
\begin_inset Formula \( t \)
\end_inset
to produce the next generation.
These are isomorphic with the truncation points
\begin_inset Formula \( x_{mt} \)
\end_inset
and density
\begin_inset Formula \( z_{mt} \)
\end_inset
of the standard normal distribution.
Breeding is by
\begin_inset Quotes eld
\end_inset
truncation
\begin_inset Quotes erd
\end_inset
: all animals better than the truncation point breed, and are randomly assigned
to another breeder.
\begin_inset Formula \( Q \)
\end_inset
is the fraction of the entire population that must be bred to produce another
population of the same size.
The initial values of
\begin_inset Formula \( p_{0} \)
\end_inset
and
\begin_inset Formula \( \bar{A}_{0} \)
\end_inset
are known.
\layout Standard
Dekkers problem was to maximize the total genetic progress by the final
generation
\begin_inset Formula \( T \)
\end_inset
:
\begin_inset Formula
\begin{equation}
\label{finobjfn}
Max_{f_{mt}}\left\{ L|\bar{A}_{0},p_{0},Q\right\}
\end{equation}
\end_inset
where
\begin_inset Formula
\begin{eqnarray}
L & = & \sum ^{T-1}_{t=0}\left\{ H_{t}-\lambda _{t}p_{t}-\gamma
_{t}\bar{A}_{t}\right\} -\lambda _{T}p_{T}+\lambda _{0}p_{0}-\gamma
_{T}\bar{A}_{0}+a\left( 2p_{T}-1\right) +\bar{A}_{T}\\
H_{t} & = & \frac{\lambda _{t+1}}{Q}\left\{ f_{1t}p_{t}^{2}+f_{2t}p_{t}\left(
1-p_{t}\right) \right\} \nonumber \\
& & +\gamma _{t+1}\left\{ \bar{A}_{t}+\frac{\sigma }{Q}\left[
p_{t}z_{1t}+2p_{t}\left( 1-p_{t}\right) z_{2t}+\left( 1-p_{t}\right) ^{2}z_{3t}\right]
\right\} \nonumber \\
& & +\epsilon _{t}\left\{ Q-f_{1t}p_{t}^{2}-2f_{2t}p_{t}\left( 1-p_{t}\right)
-f_{3t}\left( 1-p_{t}\right) ^{2}\right\}
\end{eqnarray}
\end_inset
\begin_inset LatexCommand \cite[eq 6-9]{dekkers}
\end_inset
.
\layout Standard
The extension to an infinite horizon with discounting is straightforward.
Using a constant discount value
\begin_inset Formula \( r \)
\end_inset
, note that the total genetic value in generation
\begin_inset Formula \( t \)
\end_inset
is with present value at time
\begin_inset Formula \( t=0 \)
\end_inset
is
\begin_inset Formula
\begin{equation}
\left( 1-r\right) ^{t}\bar{G}_{t}=\left( 1-r\right) ^{t}a\left( 2p_{t}-1\right)
+\bar{A}_{t}
\end{equation}
\end_inset
and the objective function to maximize becomes
\begin_inset Formula
\begin{equation}
\label{simpleinfhoriz}
G=\sum ^{\infty }_{t=0}\left( 1-r\right) ^{t}\left[ a\left( 2p_{t}-1\right)
+\bar{A}_{t}\right]
\end{equation}
\end_inset
subject to
\begin_float margin
\layout Standard
the folowing equations were broken
\end_float
\begin_inset Formula
\begin{eqnarray}
Q & = & f_{1t}p_{t}^{2}+f_{2t}2p_{t}\left( 1-p_{t}\right) +f_{3t}\left(
1-p_{t}^{2}\right) \label{siminfqcns} \\
p_{t+1} & = & \frac{1}{Q}\left\{ f_{1t}p^{2}_{t}+f_{2t}p_{t}\left( 1-p_{t}\right)
\right\} \label{siminfprule} \\
\bar{A}_{t+1} & = & \bar{A}_{t}+\frac{\sigma }{Q}\left\{ p^{2}_{t}z_{1t}+2p_{t}\left(
1-p_{t}\right) z_{2t}+\left( 1-p_{t}\right) ^{2}z_{3t}\right\} \label{siminfarule}
\end{eqnarray}
\end_inset
where (
\begin_inset LatexCommand \ref{siminfqcns}
\end_inset
) is the constraint keeping population size constant, and (
\begin_inset LatexCommand \ref{siminfprule}
\end_inset
) and (
\begin_inset LatexCommand \ref{siminfarule}
\end_inset
) describe the progression of
\begin_inset Formula \( p \)
\end_inset
and
\begin_inset Formula \( \bar{A} \)
\end_inset
given the choices made and the current state.
This formulation has the same constraints as the Dekkers formulation, save
only that the Lagrange multipliers are required for an infinite number
of time periods.
\layout Standard
However, it is useful to rewrite this exclusively in terms of
\begin_inset Formula \( x_{mt} \)
\end_inset
.
Letting
\begin_inset Formula \( \phi \)
\end_inset
and
\begin_inset Formula \( \Phi \)
\end_inset
represent the pdf and cdf, respectively, of the standard normal distribution,
\begin_inset Formula
\begin{eqnarray}
f_{mt} & = & 1-\Phi \left( x_{mt}\right) \\
z_{mt} & = & \phi \left( x_{mt}\right)
\end{eqnarray}
\end_inset
Equations (
\begin_inset LatexCommand \ref{siminfqcns}
\end_inset
-
\begin_inset LatexCommand \ref{siminfarule}
\end_inset
) become
\begin_inset Formula
\begin{eqnarray}
Q & = & \left( 1-\Phi \left( x_{1t}\right) \right) p_{t}^{2}+\left( 1-\Phi \left(
x_{2t}\right) \right) 2p_{t}\left( 1-p_{t}\right) +\left( 1-\Phi \left( x_{3t}\right)
\right) \left( 1-p_{t}^{2}\right) \label{siminfxqcons} \\
p_{t+1} & = & \frac{1}{Q}\left\{ \left( 1-\Phi \left( x_{1t}\right) \right)
p_{t}^{2}+\left( 1-\Phi \left( x_{2t}\right) \right) _{2t}p_{t}\left( 1-p_{t}\right)
\right\} \\
\bar{A}_{t+1} & = & \bar{A}_{t}+\frac{\sigma }{Q}\left\{ p_{t}^{2}\phi \left(
x_{1t}\right) +2p_{t}\left( 1-p_{t}\right) \phi \left( x_{2t}\right) +\left(
1-p_{t}\right) ^{2}\phi \left( x_{3t}\right) \right\} \\
& \forall & 0\leq t\leq \infty
\end{eqnarray}
\end_inset
The Hamiltonian remains
\begin_inset Formula
\begin{eqnarray}
H_{t} & = & \frac{\lambda _{t+1}}{Q}\left\{ \left( 1-\Phi \left( x_{1t}\right) \right)
p_{t}^{2}+\left( 1-\Phi \left( x_{2t}\right) \right) p_{t}\left( 1-p_{t}\right)
\right\} \nonumber \\
& & +\gamma _{t+1}\left\{ \bar{A}_{t}+\frac{\sigma }{Q}\left[ p_{t}^{2}\phi \left(
x_{1t}\right) +2p_{t}\left( 1-p_{t}\right) \phi \left( x_{2t}\right) +\left(
1-p_{t}\right) ^{2}\phi \left( x_{3t}\right) \right] \right\} \nonumber \\
& & +\epsilon _{t}\left\{ Q-\left( 1-\Phi \left( x_{1t}\right) \right)
p_{t}^{2}-2\left( 1-\Phi \left( x_{2t}\right) \right) p_{t}\left( 1-p_{t}\right)
-\left( 1-\Phi \left( x_{3t}\right) \right) \left( 1-p_{t}\right) ^{2}\right\}
\end{eqnarray}
\end_inset
and
\begin_inset Formula
\begin{equation}
L=G+\sum _{t=0}^{\infty }\left\{ H_{t}-\lambda _{t}p_{t}-\gamma
_{t}\bar{A}_{t}\right\} +\lambda _{0}p_{0}+\gamma _{0}\bar{A}_{0}
\end{equation}
\end_inset
\begin_float margin
\layout Standard
Up to here, I think I've made no changes.
\end_float
which differs from Dekkers' in the lack of a final period, and inclusion
of the discounted values of all generations in G.
\layout Standard
The partial derivatives of
\begin_inset Formula \( L \)
\end_inset
are taken with respect to the choice variables
\begin_inset Formula \( x_{mt} \)
\end_inset
, the state variables
\begin_inset Formula \( p_{t} \)
\end_inset
and
\begin_inset Formula \( \bar{A}_{t} \)
\end_inset
, and the Lagrangian multipliers, all of which derivatives must be equal
to zero.
\layout Standard
Taking the first partials of
\begin_inset Formula \( L \)
\end_inset
yields
\begin_inset Formula
\[
\nabla _{x_{t}}L=\nabla _{x_{t}}H\]
\end_inset
\begin_inset Formula
\begin{eqnarray}
\nabla _{x_{t}}L & = & \nabla _{x_{t}}H\nonumber \\
& = & 0\nonumber \\
& = & \frac{\lambda _{t+1}}{Q}\left[ \begin{array}{c}
-p_{t}^{2}\phi \left( x_{1t}\right) \\
-p_{t}\left( 1-p_{t}\right) \phi \left( x_{2t}\right) \\
0
\end{array}\right] \nonumber \\
& & -\frac{\gamma _{t+1}\sigma }{Q}\left[ \begin{array}{c}
p_{t}^{2}x_{1t}\phi \left( x_{1t}\right) \\
2p_{t}\left( 1-p_{t}\right) x_{2t}\phi \left( x_{2t}\right) \\
\left( 1-p_{t}^{2}\right) x_{3t}\phi \left( x_{3t}\right)
\end{array}\right] +\epsilon _{t}\left[ \begin{array}{c}
p_{t}^{2}\phi \left( x_{1t}\right) \\
2p_{t}\left( 1-p_{t}\right) \phi \left( x_{2t}\right) \\
\left( 1-p_{t}^{2}\right) \phi \left( x_{3t}\right)
\end{array}\right]
\end{eqnarray}
\end_inset
\layout Standard
For
\begin_inset Formula \( p_{t} \)
\end_inset
,
\begin_inset Formula
\begin{eqnarray}
\frac{\partial L}{\partial p_{t}} & = & 2\left( 1-r\right) ^{t}a-\lambda _{t}\nonumber
\\
& & +\frac{\lambda _{t+1}}{Q}\left\{ 2\left( 1-\Phi \left( x_{1t}\right) \right)
p_{t}+\left( 1-\Phi \left( x_{2t}\right) \right) \left( 1-2p_{t}\right) \right\}
\nonumber \\
& & +2\frac{\gamma _{t+1}\sigma }{Q}\left\{ p_{t}\phi \left( x_{1t}\right) +\left(
1-2p_{t}\right) \phi \left( x_{2t}\right) -\left( 1-p_{t}\right) \phi \left(
x_{3t}\right) \right\} \label{siminfpt} \\
& & +2\epsilon _{t}\left\{ -\left( 1-\Phi \left( x_{1t}\right) \right) p_{t}-\left(
1-\Phi \left( x_{2t}\right) \right) \left( 1-2p_{t}\right) +\left( 1-\Phi \left(
x_{3t}\right) \right) \left( 1-p_{t}\right) \right\} \nonumber
\end{eqnarray}
\end_inset
The partial with respect to
\begin_inset Formula \( \bar{A}_{t} \)
\end_inset
yields information about
\begin_inset Formula \( \gamma _{t} \)
\end_inset
, its shadow value
\begin_inset Formula
\begin{equation}
\label{simgameq}
\frac{\partial L}{\partial \bar{A}_{t}}=\left( 1-r\right) ^{t}+\gamma _{t+1}-\gamma
_{t}
\end{equation}
\end_inset
while
\begin_inset Formula \( \lambda \)
\end_inset
and
\begin_inset Formula \( \epsilon \)
\end_inset
yield only the dynamic behavior of
\begin_inset Formula \( p_{t} \)
\end_inset
and the breeding constraint:
\begin_inset Formula
\begin{eqnarray}
L_{\lambda _{t+1}} & = & \frac{1}{Q}\left\{ f_{1t}p_{t}^{2}+f_{2t}p_{t}\left(
1-p_{t}\right) \right\} -p_{t+1}\label{siminflam} \\
L_{\epsilon _{t}} & = & Q-f_{1t}p_{t}^{2}-f_{2t}2p_{t}\left( 1-p_{t}\right)
-f_{3t}\left( 1-p_{t}\right) ^{2}\label{siminfes} \\
L_{\gamma _{t+1}} & = & \bar{A}_{t}+\frac{\sigma }{Q}\left[ p_{t}^{2}\phi \left(
x_{1t}\right) +2p_{t}\left( 1-p_{t}\right) \phi \left( x_{2t}\right) +\left(
1-p_{t}\right) ^{2}\phi \left( x_{3t}\right) \right] -\bar{A}_{t+1}
\end{eqnarray}
\end_inset
Unlike the Dekkers formulation, there is no final period, and thus no derivativ
es are calculated for that special case.
\layout Standard
As all of these equations are also equal to 0,[
\begin_inset LatexCommand \ref{simgameq}
\end_inset
] means that
\begin_inset Formula
\begin{equation}
\label{siminfgamnext}
\gamma _{t+1}=\gamma _{t}-\left( 1-r\right) ^{t}
\end{equation}
\end_inset
This result is drastically different than Dekkers', which found that
\begin_inset Formula \( \gamma _{t}=1\forall t \)
\end_inset
.
As
\begin_inset Formula \( \gamma _{t} \)
\end_inset
is the shadow value for
\begin_inset Formula \( \bar{A}_{t} \)
\end_inset
,and the initial value
\begin_inset Formula \( \bar{A}_{0} \)
\end_inset
is known, it follows that
\begin_inset Formula \( \gamma _{t} \)
\end_inset
is a known value that can be directly calculated.
As the growth is additive,
\begin_inset Formula \( \bar{A}_{t} \)
\end_inset
contributes its own value in the current and each subsequent generation.
This value can be discounted to time
\begin_inset Formula \( 0 \)
\end_inset
, yielding
\begin_inset Formula
\begin{equation}
\frac{\partial G}{\partial \bar{A}_{t}}=\frac{\left( 1-r\right) ^{t}}{r}
\end{equation}
\end_inset
or that
\begin_inset Formula
\begin{equation}
\label{siminfgamval}
\gamma _{t}=\frac{\left( 1-r\right) ^{t}}{r}
\end{equation}
\end_inset
This relation can also be found by starting with the limiting value at infinity
of zero, and writing
\begin_inset Formula \( \gamma _{0} \)
\end_inset
as an infinite sum of the later values.
\layout Standard
From [
\begin_inset LatexCommand \ref{siminff1}
\end_inset
] and [
\begin_inset LatexCommand \pageref{siminfgamval}
\end_inset
] come the equations
\begin_inset Formula
\begin{equation}
\label{siminfepssln}
-\frac{\lambda _{t+1}}{Q}\left[ \begin{array}{c}
1\\
.5\\
0
\end{array}\right] +\frac{\left( 1-r\right) ^{t}\sigma }{rQ}\left[ \begin{array}{c}
x_{1t}\\
x_{2t}\\
x_{3t}
\end{array}\right] =\epsilon _{t}\left[ \begin{array}{c}
1\\
1\\
1
\end{array}\right]
\end{equation}
\end_inset
which may be solved
\begin_inset Formula
\begin{eqnarray}
x_{3t}-x_{2t} & = & \frac{r}{\left( 1-r\right) ^{t}}\frac{\lambda _{t+1}}{2\sigma
}\label{siminfx3x2} \\
x_{2t}-x_{1t} & = & \frac{r}{\left( 1-r\right) ^{t}}\frac{\lambda _{t+1}}{2\sigma
}\label{siminfx2x1}
\end{eqnarray}
\end_inset
Which gives the result that
\begin_inset Formula
\begin{equation}
x_{3t}-x_{2t}=x_{2t}-x_{1t}
\end{equation}
\end_inset
or that the truncation points are equidistant, as Dekkers found in the
finite model.
Considering the limiting case of steep discounting, this yield for the
first generation the same result as for Dekkers' final generation, as expected.
\layout Standard
The relations of [
\begin_inset LatexCommand \ref{siminfepssln}
\end_inset
] can be used to remove the
\begin_inset Formula \( \epsilon _{t} \)
\end_inset
from [
\begin_inset LatexCommand \ref{siminfpt}
\end_inset
], which becomes
\begin_inset Formula
\begin{eqnarray}
\lambda _{t} & = & 2\left( 1-r\right) ^{t}a\nonumber \\
& & +\frac{\lambda _{t+1}}{Q}\left\{ 2\left( 1-\Phi \left( x_{1t}\right) \right)
p_{t}+\left( 1-\Phi \left( x_{2t}\right) \right) \left( 1-2p_{t}\right) \right\}
\nonumber \\
& & +2\frac{\left( 1-r\right) ^{t}\sigma }{rQ}\left\{ p_{t}\phi \left( x_{1t}\right)
+\left( 1-2p_{t}\right) \phi \left( x_{2t}\right) -\left( 1-p_{t}\right) \phi \left(
x_{3t}\right) \right\} \nonumber \\
& & -\frac{\lambda _{t+1}}{Q}\left\{ 2\left( 1-\Phi \left( x_{1t}\right) \right)
p_{t}+\left( 1-\Phi \left( x_{2t}\right) \right) \left( 1-2p_{t}\right) \right\}
\nonumber \\
& & -2\frac{\left( 1-r\right) ^{t}\sigma }{rQ}\nonumber \\
& & \, \, \, \, \left\{ \left( 1-\Phi \left( x_{1t}\right) \right)
x_{1t}p_{t}+\left( 1-\Phi \left( x_{2t}\right) \right) x_{2t}\left( 1-2p_{t}\right)
-\left( 1-\Phi \left( x_{3t}\right) \right) x_{3t}\left( 1-p_{t}\right) \right\}
\nonumber \\
& = & 2\left( 1-r\right) ^{t}a+2\frac{\left( 1-r\right) ^{t}\sigma }{rQ}\left\{
\left[ \phi \left( x_{1t}\right) -\left( 1-\Phi \left( x_{1t}\right) \right)
x_{1t}\right] p_{t}\right. \nonumber \\
& & \, \, \, \, \, \left. +\left[ \phi \left( x_{2t}\right) -\left( 1-\Phi \left(
x_{2t}\right) \right) x_{2t}\right] \left( 1-2p_{t}\right) -\left[ \phi \left(
x_{3t}\right) -\left( 1-\Phi \left( x_{3t}\right) \right) x_{3t}\right] \left(
1-p_{t}\right) \right\} \label{siminfliso}
\end{eqnarray}
\end_inset
Which is an expression, if not pretty, for
\begin_inset Formula \( x_{mt} \)
\end_inset
and
\begin_inset Formula \( \lambda _{t} \)
\end_inset
.
\layout Standard
Unfortunately, this expression is not amenable to analytic or numeric solution.
For any given time period, equations [
\begin_inset LatexCommand \ref{siminfqcns}
\end_inset
], [
\begin_inset LatexCommand \ref{siminfx3x2}
\end_inset
], [
\begin_inset LatexCommand \ref{siminfx2x1}
\end_inset
], and [
\begin_inset LatexCommand \ref{siminfliso}
\end_inset
] produce five equations in only four unknowns for any given time period:
the three
\begin_inset Formula \( x_{mt} \)
\end_inset
, and the multipliers
\begin_inset Formula \( \lambda _{t} \)
\end_inset
and
\begin_inset Formula \( \lambda _{t+1} \)
\end_inset
.
If
\begin_inset Formula \( \lambda _{t} \)
\end_inset
were known for any finite period, the problem would be soluble for all
periods.
While the limiting value for large time is
\begin_inset Formula \( 0 \)
\end_inset
, namely the result that there is very little value of something that far
away, it is not possible to do an infinite summation as was done to produce
(
\begin_inset LatexCommand \ref{siminfgamval}
\end_inset
), as the remaining terms for each period are combinations of the CDF and
PDF of the standard normal distribution.
\layout Standard
The only remaining possibility would be to find a value for
\begin_inset Formula \( \lambda _{0} \)
\end_inset
.
While the expression
\begin_inset Formula
\begin{equation}
\lambda _{0}=\frac{d}{dp_{0}}G
\end{equation}
\end_inset
this is of little value, as calculating this value requires knowledge of
the solution.
\layout Standard
As such, there appears to be no solution available for the pure infinite
horizon problem.
\layout Section
Infinite Horizon With Testing Costs
\layout Standard
While the simple problem of the infinite horizon cannot be treated analytically,
due to the inability to find an initial Lagrangian multiplier for some
finite time period, the introduction of a cost for genetic testing makes
the problem tractable.
\layout Standard
Animals do not come with labels on their foreheads indicating their genetic
makeup; if so, the polygenic distribution will be known as well.
Instead, there is some finite costs for testing an animal, and testing
should not occur if the gains are outweighed by the cost.
\layout Standard
Conveniently, it is possible to put a ceiling on the value of testing.
The greatest possible gain from the major gene is for it to become fully
present in the population.
That is, for
\begin_inset Formula \( p_{t} \)
\end_inset
to change from its present value to
\begin_inset Formula \( 1 \)
\end_inset
.
Consider first the extreme case of taking this entire gain in a single
generation.
If no selection at all were to occur, the gene would keep its current frequency.
Thus the change for the next generation is
\begin_inset Formula \( 1-p_{t} \)
\end_inset
, which has a value of
\begin_inset Formula \( 2a\left( 1-p_{t}\right) \)
\end_inset
in the subsequent generation.
Further, compared to no selection, it has that value in all future generations.
Thus the present value of the change is
\begin_inset Formula
\begin{eqnarray}
PV & = & \sum _{i=1}^{\infty }\left( 1-r\right) ^{i}2a\left( 1-p_{t}\right) \nonumber
\\
& = & 2a\left( 1-p_{t}\right) \frac{1-r}{r}
\end{eqnarray}
\end_inset
Thus, in no case is it worth testing the animals if the cost of testing
is greater than this value.
\layout Standard
However, even if selection by major gene is not profitable, mass selection
would still presumably be used, allowing a tighter limit to be drawn.
Mass selection will continue to cause the frequency of the major gene to
change; animals with this gene will be selected at lower polygenic values,
and thus become relatively more common in the population.
Without testing, the observed or phenotypic truncation point will be the
same for all groups, with the result that the polygenic truncation points
are separated by exactly
\begin_inset Formula \( a \)
\end_inset
, or that
\begin_inset Formula
\begin{equation}
x_{1t}+a=x_{2t}=x_{3t}-a
\end{equation}
\end_inset
The frequency in the next generation as a function of the current frequency
can then be calculated.
First, the overall truncation point is
\begin_inset Formula
\begin{equation}
x_{t}=\Phi ^{-1}\left( 1-Q\right)
\end{equation}
\end_inset
That is, all creatures in the upper fraction
\begin_inset Formula \( Q \)
\end_inset
of the population are kept.
For creatures that don't have the gene at all, the phenotypic and polygenic
value are the same, and thus
\begin_inset Formula
\begin{equation}
x_{3t}=\Phi ^{-1}\left( 1-Q\right)
\end{equation}
\end_inset
\begin_inset Formula
\[
p_{t+1}(p_{t})=\frac{1}{Q}\left\{ \left( 1-\Phi \left( x_{2t}\right) \right) \right\}
\]
\end_inset
\layout Standard
break
\layout Standard
Given that the program of breeding for mass selection is well known, it
is possible to write a value function for the current value of the future
gains from mass selection,
\begin_inset Formula
\[
PV_{m}\left( p_{t}\right) =\sum _{i=1}^{\infty }\left( 1-r\right) ^{i}\left[ 2a\left(
p_{t+i}^{m}-p_{t}\right) +\left( \bar{A}^{m}_{t+i}-\bar{A}_{t}\right) \right] \]
\end_inset
\layout Standard
The amount of work to be invested in finding a better bound will depend
upon how much computation the bound saves; a loose bound does no harm,
but merely requires additional computation.
\layout Subsection*
Purpose of the Bound
\layout Standard
The bound has only one real purpose: if the maximu
\begin_float margin
\layout Standard
as of 2/20/99, bound has not been used
\end_float
m gain from all future testing exceeds this generation's cost of testing,
there is no reason to test, and mass selection will be used forevermore.
\layout Chapter
Genetic Algorithms
\layout Standard
Add brief discussion
\layout Chapter
Dynamic Programming
\layout Standard
Dynamic programming can only be applied to discrete problems.
However, in many cases, it is possible to discretize a continuous problem,
and find a reasonable approximation.
\layout Section
History of Dynamic Programming
\layout Standard
Dynamic programming, in and of itself, is nothing new.
It has been used as a computational method since the 1950's, but its ability
to solve problems is highly dependent upon available processing power and
fast storage (cache or core memory).
\layout Standard
Dynamic programming is used in programs that have a fixed number of available
states, and in which the
\begin_inset Quotes eld
\end_inset
history
\begin_inset Quotes erd
\end_inset
of the problem, or the path by which the state was reached, is irrelevant.
That is, the states all must have the Markov property
\begin_inset LatexCommand \index{Markov property}
\end_inset
.
\latex latex
\backslash
jim{
\latex default
A state is "Markov" if the state contains all relevant information, regardless
of how it was reached.
If the state has the Markov property, it does not matter in which generation
the state was reache, nor does the path by which it was reached matter.
\latex latex
}
\begin_float margin
\layout Standard
more on Markov?
\end_float
\layout Standard
Dynamic programming can thus be used to solve policy questions, such as
the order of applying treatment to fields, in which both
\emph on
which
\emph default
steps to take as well as which
\emph on
order
\emph default
to take them in are considered.
The value of each state is known, and therefore once both steps of the
tentative solution
\begin_inset Quotes eld
\end_inset
apply forty tons of phosphor, then plant corn
\begin_inset Quotes erd
\end_inset
is calculated, considering the solution,
\begin_inset Quotes eld
\end_inset
apply twenty tons of herbicide, then forty tons of phosphor, then plant
corn
\begin_inset Quotes erd
\end_inset
requires only calculating the effect of the first step, and then using
the already calculated value for corn.
\layout Standard
While the problems soluble by dynamic programming are inherently discrete,
there is a long history of its use in solving continuous problems.
This is done by the expedient of discretizing the problem, allowing only
certain states, a procedure that dates to **.
\begin_float margin
\layout Standard
get this
\end_float
In some cases, this approximate solution is sufficient.
For example, if the problem involves a
\begin_inset Formula \( [0,1] \)
\end_inset
choice variable, and knowledge of the solution to within
\begin_inset Formula \( .01 \)
\end_inset
is sufficient, then only
\begin_inset Formula \( 100 \)
\end_inset
states need be considered.
If such an answer is not precise enough, the first solution can be used
as a center for a search over a smaller area with a finer grid.
\layout Standard
However, published numeric solutions almost universally consider compact
search spaces.
The few examples of non-compact spaces
\begin_float margin
\layout Standard
ref
\end_float
have all used contiguous spaces, which are not appropriate for the breeding
problem.
\layout Section
The Breeding Problem and Dynamic Programming
\layout Standard
The naive approach to the breeding problem would be to look at the solution
spaces as
\begin_inset Formula \( \left\{ f_{mt}\right\} \)
\end_inset
.
However, this is far too complicated a space to use as a state space for
dynamic programming: at even a resolution of
\begin_inset Formula \( .01 \)
\end_inset
, there are
\begin_inset Formula \( 100 \)
\end_inset
possible values for each choice, or
\begin_inset Formula \( 10^{4} \)
\end_inset
per generation with only one choice gene.
Then for
\begin_inset Formula \( T \)
\end_inset
generations, there are
\begin_inset Formula \( 10^{4T} \)
\end_inset
possible states to consider.
\layout Standard
However, for a finite horizon problem, or an infinite horizon problem with
a known
\begin_inset Formula \( \hat{p} \)
\end_inset
bound such as [
\begin_float margin
\layout Standard
ref
\end_float
],
\begin_inset Formula \( p_{t} \)
\end_inset
alone may be used as the state.
\begin_inset Formula \( p \)
\end_inset
may be divided into as many states as desired, and and the optimal choice
for each value of
\begin_inset Formula \( p \)
\end_inset
can be calculated.
\layout Standard
Consider again the nature of the general problem: once a change has been
made to
\begin_inset Formula \( \bar{A}_{t} \)
\end_inset
, the change is permanent.
The only choice to be made depends upon the present value of
\begin_inset Formula \( p_{t} \)
\end_inset
.
\begin_float margin
\layout Standard
Will non-constant variance destroy this?
\end_float
The optimal behavior for
\begin_inset Formula \( p_{t}>\hat{p} \)
\end_inset
is known, namely to switch to mass selection.
This is entered into the action space as a beginning.
Letting
\begin_inset Formula
\begin{equation}
\Delta =\frac{1}{S}
\end{equation}
\end_inset
where
\begin_inset Formula \( \Delta \)
\end_inset
is the spacing between potential values of
\begin_inset Formula \( p_{t} \)
\end_inset
for a number of states
\begin_inset Formula \( S \)
\end_inset
, the first action considered is for
\begin_inset Formula \( p_{t}=\hat{p}-\Delta \)
\end_inset
.
There are only two choice variables to consider,
\begin_inset Formula \( f_{1t} \)
\end_inset
and
\begin_inset Formula \( f_{2t} \)
\end_inset
.
As
\begin_inset Formula \( p_{t+1}\geq p_{t} \)
\end_inset
, for each value of of the fractions considered, the future path is known.
That is, for each trial value of
\begin_inset Formula \( f_{t} \)
\end_inset
considered,
\begin_inset Formula \( p_{t+1}\left( f_{t}\right) \)
\end_inset
is calculated, and the future value of this choice is selected from the
table of known results.
If this value is less than testing costs, the decision not to test is stored
for this variable, a is the present value of mass selection at this frequency.
Additionally, a tighter bound has been found, and
\begin_inset Formula \( \hat{p} \)
\end_inset
takes on this new value.
If testing was worthwhile, this fact is stored, as testing will be worthwhile
at all smaller frequencies,
\begin_float margin
\layout Standard
prove this?
\end_float
as are the optimal fractions and the net present value of this frequency.
\layout Standard
This step is repeated until a frequency is found which finds value in testing,
at which point the comparison to testing cost is skipped for all smaller
values, and
\begin_inset Formula \( \hat{p} \)
\end_inset
is reset to
\begin_inset Formula \( \Delta \)
\end_inset
greater than this frequency.
Actions for
\begin_inset Formula \( p_{t} \)
\end_inset
are then calculated for all smaller values.
\layout Standard
Eventually, values of
\begin_inset Formula \( p_{t} \)
\end_inset
will be considered which find
\begin_inset Formula \( p_{t+1}<\hat{p} \)
\end_inset
for the optimal values of
\begin_inset Formula \( f_{t} \)
\end_inset
.
This is easily handled; for each trial value of
\begin_inset Formula \( f_{t} \)
\end_inset
, the value of the resultant
\begin_inset Formula \( p_{t+1} \)
\end_inset
has already been saved, and need not be recalculated.
\layout Standard
For a single state variable
\begin_inset Formula \( p_{t} \)
\end_inset
, this appears to be an efficient search: rather than checking all possible
values of
\begin_inset Formula \( p_{t} \)
\end_inset
, only a portion are checked.
While this is convenient for the single state variable, it will be critical
as the number of state variables increases.
\layout Section
An Initial Algorithm
\layout Standard
Consider first the simple problem of one gene with fixed variance.
For any given
\begin_inset Formula \( p_{t} \)
\end_inset
and
\begin_inset Formula \( \bar{A}_{t} \)
\end_inset
, there is an associated maximal present value
\begin_inset Formula \( G\left( p_{t}\right) \)
\end_inset
.
First note that the starting value
\begin_inset Formula \( \bar{A}_{t} \)
\end_inset
affects the present value by only a known offset
\begin_inset Formula
\begin{equation}
G\left( p_{t},\bar{A}_{t}\right) =G\left( p_{t},0\right) +\frac{\bar{A}_{t}}{r}
\end{equation}
\end_inset
Then break
\begin_inset Formula \( G \)
\end_inset
into three pieces: the initial fixed effects, which choices in
\begin_inset Formula \( t \)
\end_inset
cannot affect, the initial effects in the subsequent generation, and all
time after that:
\begin_inset Formula
\begin{eqnarray*}
G\left( p_{t}\right) & = & \bar{A}_{t}+a\left( 2p_{t}-1\right) \\
& & +\left( 1-r\right) \left[ \frac{\sigma }{Qr}\left\{
2p_{t+1}^{2}z_{1t+1}+2p_{t+1}\left( 1-p_{t+1}\right) z_{2t+2}+\left( 1-p_{t+1}\right)
^{2}z_{3t+1}\right\} +\bar{A}_{t}\right] \\
& & +\left( 1-r\right) a\frac{1}{Q}\left\{ 2f_{1t}p_{t}^{2}+f_{2t}p_{t}\left(
1-2p_{t}\right) \right\} \\
& & +\sum _{s=t+2}^{\infty }\left( 1-r\right) ^{s}\left[ a\left( 2p_{t}-1\right)
+\bar{A}_{s}\right]
\end{eqnarray*}
\end_inset
Note that part of the value of
\begin_inset Formula \( \bar{A}_{s} \)
\end_inset
in the summation can be regrouped into the first term, as can the appearance
of
\begin_inset Formula \( \bar{A}_{t} \)
\end_inset
in the period
\begin_inset Formula \( t+1 \)
\end_inset
.
However, another approach will be more fruitful, namely redefining the
objective function.
Consider
\begin_inset Formula
\begin{eqnarray}
F\left( p_{t}\right) & \equiv & +\frac{\sigma }{Qr}\left\{
2p_{t}^{2}z_{1t}+2p_{t}\left( 1-p_{t}\right) z_{2t}+\left( 1-p_{t}\right)
^{2}z_{3t}\right\} \nonumber \\
& & +\frac{a}{Q}\left\{ 2f_{1t}p_{t}^{2}+f_{2t}p_{t}\left( 1-2p_{t}\right) \right\}
\nonumber \\
& & +\left( 1-r\right) F\left( p_{t+1}\right)
\end{eqnarray}
\end_inset
As
\begin_inset Formula
\begin{equation}
G\left( p_{t}\right) =\frac{\bar{A}_{t}}{r}+a\left( 2p_{t}-1\right) +\left( 1-r\right)
F\left( p_{t+1}\left( p_{t}\right) \right)
\end{equation}
\end_inset
maximizing
\begin_inset Formula \( F \)
\end_inset
homomorphous with maximizing
\begin_inset Formula \( G \)
\end_inset
.
\layout Standard
The problem is now in a state to which dynamic programming can be applied.
Using
\begin_inset Formula \( S \)
\end_inset
possible states for
\begin_inset Formula \( p_{t+1} \)
\end_inset
, let
\begin_inset Formula \( p_{t+1}\left( p_{t},f_{t}\right) \)
\end_inset
be the closest allowed state to the actual computed value.
Finding optimal values for
\begin_inset Formula \( f_{t}\left( p_{t}\right) \)
\end_inset
becomes almost trivial.
\layout Standard
The existence of
\begin_inset Formula \( F(p_{t+1>t}) \)
\end_inset
is taken as a given; it will be called recursively if necessary.
This reduces the problem to finding the best set of
\begin_inset Formula \( f_{t} \)
\end_inset
for the current generation.
An initial trial solution is considered, from whatever source, yielding
a tentative
\begin_inset Formula \( p^{t}_{t+1} \)
\end_inset
.
\begin_inset Formula \( F\left( p_{t+1}^{t}\right) \)
\end_inset
is treated as known, and the remainder of
\begin_inset Formula \( F\left( p_{t}\right) \)
\end_inset
is optimized over
\begin_inset Formula \( \left\{ f_{t}:p_{t+1}\left( p_{t},f_{t}\right)
=p_{t+1}^{t}\right\} \)
\end_inset
, for which standard optimization methods suffice.
This is compared to the maximum over
\begin_inset Formula \( \left\{ f_{t}:p_{t+1}\left( p_{t},f_{t}\right)
=p_{t+1}^{t}+\delta \right\} \)
\end_inset
, and the better value is stored.
\layout Standard
The optimal value and behavior with granularity
\begin_inset Formula \( \delta \)
\end_inset
can now be found by simply calling this optimal value function for
\begin_inset Formula \( F\left( p_{0}\right) \)
\end_inset
, which calls itself recursively to find any other needed values for other
values of
\begin_inset Formula \( p_{t} \)
\end_inset
.
Many possible states will likely be passed over, saving computational time.
These solutions can then be used as starting values for another run with
a finer grain, until the desired level of resolution is found.
\layout Standard
As a side effect of this method, the stiff matrix problem found with the
genetic algorithm in section [] is avoided entirely: there is never an
attempt to directly determine the present value of the effect of a change
in a current variable on a state several generations away; all such calculation
s find the optimal value by looking ahead only one generation.
\layout Section
Application to Multiple State Variables
\layout Standard
The method does not translate directly to multiple variables, although adaptatio
ns are possible.
Similar to the problem at [], the search space explodes as additional states
are considered.
For example, with a grain of
\begin_inset Formula \( .01 \)
\end_inset
, only
\begin_inset Formula \( 101 \)
\end_inset
states exist for the simple model of section [].
However, adding only a second gene, still keeping the variance fixed, increases
this toe
\begin_inset Formula \( 101^{2} \)
\end_inset
.
Adding variance for each gene, and covariance, reaches
\begin_inset Formula \( 101^{5} \)
\end_inset
, or in excess of ten billion states to consider.
Storing a single double precision floating point variable for each of these
would take on the order of eighty gigabytes of temporary storage.
\layout Standard
The above method, however, allowed the possibility of all possible combinations
of allowed values.
However, if there is a
\begin_inset Quotes eld
\end_inset
reasonably good
\begin_inset Quotes erd
\end_inset
starting point, this is not necessary.
Instead, from this starting point, only a small range of values need be
considered.
\layout Standard
For example, suppose that consideration of of the mass selection solution
allows placing a bound of fifteen generations, and provides initial values
for
\begin_inset Formula \( \left\{ p_{t}^{1},p_{t}^{2},\sigma _{t}^{1},\sigma
_{t}^{2},\sigma _{t}^{12}:0\leq t\leq 15\right\} \)
\end_inset
.
Suppose further that the researcher is experienced with the problem, perhaps
from prior runs, and is thus able to set likely bounds for the values in
each generation.
While it is not possible to store or even allocate space for all possible
values in the domain, it may be possible to create
\begin_inset Quotes eld
\end_inset
disjoint bubbles
\begin_inset Quotes erd
\end_inset
within the space for examination.
\layout Standard
For graphical simplicity, consider a simple case of two probabilities and
their correlation.
Starting with low values for each of the three, it seems reasonable to
expect that each of the probabilities and the correlation will approach
\begin_inset Formula \( 1 \)
\end_inset
in the face of selection.
With an initial trial solution as in figure (
\begin_inset LatexCommand \ref{bubblelines}
\end_inset
),
\begin_float fig
\layout Standard
\align center
\begin_inset Figure size 133 184
file graphics/bubblelines.eps
width 4 45
angle 270
flags 9
\end_inset
\layout Caption
\begin_inset LatexCommand \label{bubblelines}
\end_inset
Initial Trial Solution
\end_float
and a grain of
\begin_inset Formula \( .01 \)
\end_inset
for each of the three variables, there are
\begin_inset Formula \( 10^{6} \)
\end_inset
possible states, a manageable number.
However, the researcher is likely to desire a greater granularity,
\begin_inset Formula \( .0001 \)
\end_inset
, or even
\begin_inset Formula \( .000001 \)
\end_inset
, requiring
\begin_inset Formula \( 10^{12} \)
\end_inset
or
\begin_inset Formula \( 10^{18} \)
\end_inset
states, respectively.
\layout Standard
\pextra_type 3
However, the changes in each variable from period to period are typically
far larger than the grain; there are large ranges in each variable which
are known ahead of time to be unlikely to be reached.
Instead, some region, a cube in
\begin_inset Formula \( \Re ^{3} \)
\end_inset
around each point of the trial solution is most likely to be reached figure
(
\begin_inset LatexCommand \ref{bubbles1}
\end_inset
).
\layout Standard
\begin_float fig
\layout Standard
\align center
\begin_inset Figure size 133 184
file graphics/bubbles1.eps
width 4 45
angle 270
flags 9
\end_inset
\layout Caption
\begin_inset LatexCommand \label{bubbles1}
\end_inset
Partition of space near trial solution into disjoint bubblettes.
\end_float
While there is no
\emph on
a priori
\emph default
reason to assume a cube (or hypercube in higher dimensions), this cube
can be chosen so as to include any size and shape of region.
\layout Standard
\pextra_type 3 \pextra_widthp 60
If reasonable
\emph on
a priori
\emph default
bounds can be placed on the size of these cubes for each step, and this
region is divided into ten possible values for each of the three variables,
each such bubble contains only
\begin_inset Formula \( 10^{3} \)
\end_inset
states, which is manageable for even large number of generations.
Figure (
\begin_inset LatexCommand \ref{bubbles2}
\end_inset
)shows five possible states for each state variable.
If the trial solution is
\begin_inset Quotes eld
\end_inset
close enough,
\begin_inset Quotes erd
\end_inset
or if the bubbles are large enough, the solution at each point will be
within one of the divisions of the cube, and the dynamic programming approach
would then seem to be successive steps with finer grain and smaller bubbles
in each period.
For example, if the solution shown in figure(
\begin_inset LatexCommand \ref{bubbles3}
\end_inset
)is found to be the best, smaller bubbles and bubblettes may be found around
this new tentative solution, as in figure (
\begin_inset LatexCommand \ref{bubbles4}
\end_inset
)This process would be repeated until with sufficiently fine grain the solution
does not move.
\begin_float fig
\layout Standard
\align center
\begin_inset Figure size 133 184
file graphics/bubbles2.eps
width 4 45
angle 270
flags 9
\end_inset
\layout Caption
\begin_inset LatexCommand \label{bubbles2}
\end_inset
Division of bubbles into bubblettes
\end_float
\begin_float fig
\layout Standard
\align center
\begin_inset Figure size 133 184
file graphics/bubbles3.eps
width 4 45
angle 270
flags 9
\end_inset
\layout Caption
\begin_inset LatexCommand \label{bubbles3}
\end_inset
Possible new trajectory
\end_float
\begin_float fig
\layout Standard
\align center
\begin_inset Figure size 133 184
file graphics/bubbles4.eps
width 4 45
angle 270
flags 9
\end_inset
\layout Caption
\begin_inset LatexCommand \label{bubbles4}
\end_inset
New disjoint bubbles and bubblettes
\end_float
\layout Subsection
Missing the Bubbles
\layout Standard
The foregoing example has a critical implicit assumption: that the number
of steps to the solution is known.
This is important as the bubbles are identified with with a specific step.
It is entirely possible that these bubbles overlap; this is not a problem.
The problem, rather, is that that a tentative solution steps outside of
the
\begin_inset Quotes eld
\end_inset
next
\begin_inset Quotes erd
\end_inset
bubble, as in figure (
\begin_inset LatexCommand \ref{bubblemiss}
\end_inset
).
\begin_float fig
\layout Standard
\align center
\begin_inset Figure size 133 191
file graphics/bubblemiss.eps
width 4 45
angle 270
flags 9
\end_inset
\layout Caption
\begin_inset LatexCommand \label{bubblemiss}
\end_inset
Missing the bubbles entirely
\end_float
To this point, it has been assumed that the each step will land in a known
neighborhood, that of the bubble previously associated with that step.
However, the dashed line lands in the
\begin_inset Quotes eld
\end_inset
wrong
\begin_inset Quotes erd
\end_inset
bubble, while the dotted line fails to land within
\emph on
any
\emph default
bubble.
\layout Standard
This is not a trivial problem.
The essence of dynamic programming is to reuse previously calculated states.
However, the bubbles have been used because the number of potential states
is too great to store, let alone calculate.
If a step is made outside a bubble, it would seem likely that the next
step is also outside the bubble.
Each of these will require calculations of multiple possible states for
comparison.
Some manner must be found to index or search through the bubbles, so that
once a new bubble has been created, later steps can find it and take advantage
of its calculations.
\layout Standard
Another source of misstep can come with a change in the number of generations
in the solution.
For example, if the initial solution takes fifteen generations before reaching
a state in which testing is not profitable, it may be that a path is found
which reaches this level in fourteen generations.
This may mean that the thirteenth step proceeds to the bubble previously
associated with the fifteenth, as with the dashed line in figure (
\begin_inset LatexCommand \ref{bubblemiss}
\end_inset
).
\layout Section
A Structure for the Bubblettes
\layout Standard
Before turning to the Bubbles, it is necessary to consider the nature of
the bubblettes.
\layout Standard
Most fundamentally, each bubblettes has three possible states: fully calculated,
fully uncalculated, and uncalculated with hints of some nature.
\layout Standard
The fully calculated bubblette must, at a minimum, contain the following
information: the fact that it is calculated, all state variables for the
bubblette, and the
\begin_inset Quotes eld
\end_inset
next
\begin_inset Quotes erd
\end_inset
bubblette--the bubblette reached in the next generation.
However, more information is desirable, and eases calculation.
Particularly, it is desirable to store the choice variables, such as the
\begin_inset Formula \( f_{t} \)
\end_inset
, which lead to the next bubblette.
Additionally, the value of the objective function for the bubblette is
desirable--it allows a later bubblette which considers this state to do
so by making a single stop, rather than considering the successor states.
Finally, some
\begin_inset Quotes eld
\end_inset
hint
\begin_inset Quotes erd
\end_inset
as to the location of the next bubblette may be desirable; it is not clear
that knowledge of the next bubblette necessarily means where it is located
within the machine.
\layout Standard
If a bubble is not yet calculated, it may yet have a
\begin_inset Quotes eld
\end_inset
hint
\begin_inset Quotes erd
\end_inset
left from a prior run or some other information.
This hint, perhaps the result for this or an adjacent location in a prior
run, can be used as a starting point when the bubblette searches for its
values.
However, storing the hint does take storage space.
As such, it may be desirable to add a pointer variable to bubblettes for
hints, allowing them to point to another bubblette for a hint, rather than
storing information itself.
\layout Section
A structure for the bubbles
\layout Standard
As the bubble is divided into bubblettes, it clearly must, at a minimum,
contain storage space for it's constituent bubblettes.
While it is conventional to think of
\begin_inset Quotes eld
\end_inset
balls
\begin_inset Quotes erd
\end_inset
in n-space for such a region, the hypercube is more natural to the computer:
a
\begin_inset Formula \( 5\times 5\times \ldots \times 5 \)
\end_inset
region is well defined in memory, but the set of all regions within distance
\begin_inset Formula \( 5 \)
\end_inset
is not.
The result is probably wasted states, but it may be possible to mitigate
this by storing not bubblettes, but pointers to bubblettes, and only allocating
space for bubblettes as needed.
For eleven possible values for each of five state variables, this means
\begin_inset Formula \( 161051 \)
\end_inset
pointers, which will take
\begin_inset Formula \( 629 \)
\end_inset
kilobytes of storage on a thirty-two bit machine.
While this is a massive amount of storage, and will be largely unused,
this is still less than ten megabytes for fifteen bubbles, leaving the
balance for calculated states.
\layout Standard
It will generally be assumed that there are an odd number of levels for
each variable: the center of the bubble comes from some prior knowledge
or solution, and a symmetric number of levels on each side results in an
odd number.
\layout Standard
Merely containing the bubblettes, however, is not enough for the bubble
structure.
It is necessary to search the bubbles, so that they are not needlessly
duplicated.
Therefore, each bubble should store the domain for each state variable.
To facilitate a rapid search, it may be desirable to have these actually
be pointers elsewhere; arrays of minima and maxima for each state variable
can be kept elsewhere, with a pointer stored in the bubble structure.
A stranded step could then search for an existing bubble rapidly with a
construct such as
\layout LyX-Code
where ( (mystate1.ge.state1min) .and.
(mystate1 .le.state1max) .and.
(mystate2.ge.state2min) ...)
\layout LyX-Code
\protected_separator
\protected_separator
candidateBubbles=.true.
\layout LyX-Code
elsewhere
\layout LyX-Code
\protected_separator
\protected_separator
candidateBubbles=.false.
\layout LyX-Code
endwhere
\layout Standard
thereby gaining a list of candidate bubble instantly.
By comparison, if it had to reference the variables within structures to
do comparisons, the search would be slowed by orders of magnitude.
If a bubble is found, the bubblette is either calculated or referenced.
If not, a new bubble is created, with the state in question at its center.
\layout Standard
Another useful feature in a bubble would be reference hints for uncalculated
bubblettes.
After a run of the algorithm, bubblettes will exist for some or all of
the bubblettes, many of which will be contained within the corresponding
bubble on the next run.
These values can be saved as hints for the corresponding bubblettes within
the smaller bubbles of the next pass.
\layout Section
An initial strategy
\layout Standard
At this stage, it is possible to make an initial attempt to use dynamic
programming to solve a problem.
From the performance of the attempt, information will be gained for later
refinement.
\layout Subsection
Replacing choice variables with state variables
\layout Standard
\begin_inset LatexCommand \label{sec:replacechoicevars}
\end_inset
One final obstacle must be addressed before attempting a dynamic programming
solution: the discretization of the choice space.
\layout Standard
While the state space, and permitted tentative state spaces, have already
been defined in terms of bubbles in [], these states are reached as result
of tentative selection of choice variables.
Furthermore, the literature on discretized problems has generally assumed
the discretization of the choice variables.
However, discretizing the
\begin_inset Formula \( f_{t} \)
\end_inset
variables in this case will not result in a cleanly indexable space for
the state variables, and the success of any dynamic programming solution
relies on the ability to easily look up prior solved situations.
Finally, there is no
\emph on
a priori
\emph default
reason to believe that such a discretization of the choice variables would
result in a state space in which values would be repeated.
\begin_float margin
\layout Standard
Maybe define a
\begin_inset Quotes eld
\end_inset
close enough
\begin_inset Quotes erd
\end_inset
in state space as an alternative approach? But the search costs will be
staggering.
\end_float
\layout Standard
Another approach would lie
\begin_float margin
\layout Standard
it is
\begin_inset Quotes eld
\end_inset
lie,
\begin_inset Quotes erd
\end_inset
isn't it?
\end_float
in keeping the discretization of state space, while fixing
\begin_inset Quotes eld
\end_inset
canonical
\begin_inset Quotes erd
\end_inset
values for allowed choices.
As search algorithms generally use some type of Newton method to select
trial solutions, it is necessary to be able to define the distance between
points.
Consider, for example, the function
\begin_inset Formula
\begin{equation}
y=-\left( x-3\right) ^{4}
\end{equation}
\end_inset
ignoring for the moment that a trivial inverse exists.
This function takes a maximum value of
\begin_inset Formula \( 0 \)
\end_inset
at
\begin_inset Formula \( x=3 \)
\end_inset
.
Suppose that the search range has been reduced to
\begin_inset Formula \( \left[ -0.750,.250\right] \)
\end_inset
, with a grain of
\begin_inset Formula \( .001 \)
\end_inset
for
\begin_inset Formula \( x \)
\end_inset
, centered around the initial guess of
\layout Standard
\begin_inset Formula
\begin{eqnarray}
\hat{x} & = & 3.3\nonumber \\
\hat{y} & = & -.008
\end{eqnarray}
\end_inset
Calculating numerical derivatives becomes problematic.
The actual derivative is
\begin_inset Formula
\begin{equation}
\frac{dy}{dx}=-4\left( x-3\right) ^{3}
\end{equation}
\end_inset
so that to change
\begin_inset Formula \( y \)
\end_inset
by
\begin_inset Formula \( .001 \)
\end_inset
requires a change of approximately
\begin_inset Formula \( .00926 \)
\end_inset
.
However, changes of
\begin_inset Formula \( .007 \)
\end_inset
and
\begin_inset Formula \( .014 \)
\end_inset
will both result in a calculated
\begin_inset Formula \( y \)
\end_inset
of
\begin_inset Formula \( .007 \)
\end_inset
, though the numerical derivative of the former is twice the size of the
latter.
Numerical calculation of the Hessian would result in even worse exaggerations.
\layout Standard
To solve this problem would require limiting the allowed values of
\begin_inset Formula \( x \)
\end_inset
, so that for all
\begin_inset Formula \( \left\{ x:\hat{f}\left( x\right) =\hat{y}\right\} \)
\end_inset
,
\begin_inset Formula \( f\left( \hat{x}(x)\right) = \)
\end_inset
\begin_inset Formula \( \hat{y} \)
\end_inset
, and to work only with
\begin_inset Formula \( \hat{x} \)
\end_inset
and not
\begin_inset Formula \( x \)
\end_inset
itself.
\layout Standard
However, as dimensions increase, or as functions which are not invertible
inexpensively are used, this will become an impossible problem.
\layout Standard
These difficulties suggest that transforming the problem into one of choosing
state spaces transitions from those permitted, and afterwards transforming
the solution back into the choices that reach them, may be the easiest
way to solve the problem.
Given rules on state space transitions, such as Equations [], the back-conversi
on should run in linear time in the number of generations used.
\begin_float margin
\layout Standard
more on this?
\end_float
\layout Subsection
Simple Dynamic Programming Model
\layout Standard
For an initial model to demonstrate the concept, consider again the simple
model of [
\begin_inset LatexCommand \ref{finobjfn}
\end_inset
].
As this model has a known solution, it will be the first to be solved.
However, in designing structures and methods, more attention will be paid
to the usefulness of the model in solving the general case than the problem
at hand.
\layout Standard
Using equations [
\begin_inset LatexCommand \ref{siminfprule}
\end_inset
\begin_inset LatexCommand \ref{siminfarule}
\end_inset
\begin_inset LatexCommand \ref{siminfqcns}
\end_inset
],
\begin_inset Formula
\begin{eqnarray}
p_{t+1} & = & \frac{1}{Q}\left\{ f_{1t}p_{t}^{2}+f_{2t}p_{t}\left( 1-p_{t}\right)
\right\} \label{dp2prule} \\
\bar{A}_{t+1} & = & \bar{A}_{t}+\frac{\sigma }{Q}\left\{ p_{t}^{2}z_{1t}+2p_{t}\left(
1-p_{t}\right) z_{2t}+\left( 1-p_{t}^{2}\right) z_{3t}\right\} \label{dp2abarrule} \\
Q & = & f_{1t}p_{t}^{2}+f_{2t}2p_{t}\left( 1-p_{t}\right) +f_{3t}\left(
1-p_{t}^{2}\right) \label{dp2qcons}
\end{eqnarray}
\end_inset
[
\begin_inset LatexCommand \ref{siminfprule}
\end_inset
] can be manipulated into
\begin_inset Formula
\begin{equation}
\label{dpsimf2off1}
f_{2t}\left( f_{1t}|p_{t+1}\right) =\frac{Qp_{t+1}-f_{1t}p_{t}^{2}}{p_{t}\left(
1-p_{t}\right) }
\end{equation}
\end_inset
[
\begin_inset LatexCommand \ref{siminfqcns}
\end_inset
] becomes
\begin_inset Formula
\begin{eqnarray}
f_{3t}\left( f_{1t}|p_{t+1}\right) & = & \frac{Q-f_{1t}p_{t}^{2}-f_{2t}\left(
f_{1t}|p_{t+1}\right) 2p_{t}\left( 1-p_{t}\right) }{\left( 1-p_{t}\right)
^{2}}\nonumber \\
& = & \frac{Q-f_{1t}p_{t}^{2}-\frac{Qp_{t+1}-f_{1t}p_{t}^{2}}{p_{t}\left(
1-p_{t}\right) }2p_{t}\left( 1-p_{t}\right) }{\left( 1-p_{t}\right) ^{2}}\nonumber
\label{dpsimf3off1} \\
& = & \frac{Q+f_{1t}p_{t}^{2}-2Qp_{t+1}}{\left( 1-p_{t}\right)
^{2}}\label{dpsimf3off1}
\end{eqnarray}
\end_inset
and [
\begin_inset LatexCommand \ref{siminfarule}
\end_inset
] becomes
\begin_inset Formula
\begin{equation}
\label{dpsiminfanextoff1}
\bar{A}_{t+1}=\bar{A}_{t}+\frac{\sigma }{Q}\left\{ \begin{array}{c}
p_{t}^{2}\phi \left( \Phi ^{-1}\left( 1-f_{1t}\right) \right) \\
+2p_{t}\left( 1-p_{t}\right) \phi \left( \Phi ^{-1}\left(
+1-\frac{Qp_{t+1}-f_{1t}p_{t}^{2}}{p_{t}\left( 1-p_{t}\right) }\right) \right) \\
+\left( 1-p_{t}^{2}\right) \phi \left( \Phi ^{-1}\left(
+1-\frac{Q+f_{1t}p_{t}^{2}-2Qp_{t+1}}{\left( 1-p_{t}\right) ^{2}}\right) \right)
\end{array}\right\}
\end{equation}
\end_inset
\layout Standard
This allows the translation of the problem into a one-dimensional problem
for purposes of dynamic programming.
The sole variable is
\begin_inset Formula \( p_{t+1} \)
\end_inset
, and [
\begin_inset LatexCommand \ref{dpsiminfanextoff1}
\end_inset
] is used to find the maximal
\begin_inset Formula \( \bar{A}_{t+1} \)
\end_inset
for the chosen
\begin_inset Formula \( p_{t+1} \)
\end_inset
.
To make the calculation useful for dynamic programming, it is actually
the change, the maximal value of
\begin_inset Formula \( \bar{A}_{t+1}-\bar{A}_{t} \)
\end_inset
that is calculated, so that the information is useful regardless of the
prior state.
Also, given that the available libraries minimize rather than maximize,
and that
\begin_inset Formula \( \bar{A}_{t+1} \)
\end_inset
is linear in
\begin_inset Formula \( Q \)
\end_inset
and
\begin_inset Formula \( p \)
\end_inset
, the new function
\begin_inset Formula \( y \)
\end_inset
is defined as
\layout Standard
\begin_inset Formula
\begin{eqnarray}
y & = & -\frac{Q}{p}\left( \bar{A}_{t+1}-\bar{A}_{t}\right) \nonumber \\
& = & -p_{t}^{2}z_{1t}-2p_{t}\left( 1-p_{t}\right) z_{2t}-\left( 1-p_{t}\right)
^{2}z_{3t}\nonumber \\
& = & -p_{t}^{2}z_{1t}-2p_{t}\left( 1-p_{t}\right) z\left(
\frac{Qp_{t+1}-f_{1t}p_{t}^{2}}{p_{t}\left( 1-p_{t}\right) }\right) \nonumber \\
& & -\left( 1-p_{t}\right) ^{2}z\left( \frac{Q+f_{1t}p_{t}^{2}-2Qp_{t+1}}{\left(
1-p_{t}\right) ^{2}}\right)
\end{eqnarray}
\end_inset
with the derivative
\begin_inset Formula
\begin{eqnarray}
\frac{dy}{df_{1t}} & = & -p_{t}^{2}x\left( f_{1t}\right) -2p_{t}\left( 1-p_{t}\right)
x\left( f_{2t}\left( f_{1t}\right) \right) \frac{df_{2t}}{df_{1t}}\nonumber \\
& & -\left( 1-p_{t}\right) ^{2}x\left( f_{3t}\left( f_{1t}\right) \right)
\frac{df_{3t}}{df_{1t}}\nonumber \\
& = & -p_{t}^{2}\Phi ^{-1}\left( 1-f_{1t}\right) -2p_{t}\left( 1-p_{t}\right) \Phi
^{-1}\left( 1-f_{2t}\left( f_{1t}\right) \right) \frac{-p_{t}^{2}}{p_{t}\left(
1-p_{t}\right) }\nonumber \\
& & -\left( 1-p_{t}\right) ^{2}\Phi ^{-1}\left( 1-f_{3t}\left( f_{1t}\right) \right)
\frac{p_{t}^{2}}{\left( 1-p_{t}\right) ^{2}}\nonumber \\
& = & -p_{t}^{2}\Phi ^{-1}\left( 1-f_{1t}\right) +2p_{t}^{2}\Phi ^{-1}\left(
1-f_{2t}\right) \nonumber \\
& & -p_{t}^{2}\Phi ^{-1}\left( 1-f_{3t}\right)
\end{eqnarray}
\end_inset
\layout Standard
or?
\begin_inset Formula
\begin{eqnarray}
\frac{dy}{df_{1t}} & = & -p_{t}^{2}x_{mt}\left( f_{1t}\right) -2p_{t}\left(
1-p_{t}\right) x\left( \frac{Qp_{t+1}-f_{1t}p_{t}^{2}}{p_{t}\left( 1-p_{t}\right)
}\right) \frac{df_{2t}}{df_{1t}}\nonumber \\
& & -\left( 1-p_{t}\right) ^{2}x\left(
\frac{Q-f_{1t}p_{1t}^{2}-\frac{Qp_{t+1}-f_{1t}p_{t}^{2}}{p_{t}\left( 1-p_{t}\right)
}}{\left( 1-p_{t}\right) ^{2}}\right) \frac{df_{3t}}{df_{1t}}\nonumber \\
& = & -p_{t}^{2}\Phi ^{-1}\left( f_{1t}\right) -2p_{t}^{2}\Phi ^{-1}\left(
\frac{Qp_{t+1}-f_{1t}p_{t}^{2}}{p_{t}\left( 1-p_{t}\right) }\right)
\frac{-p_{t}^{2}}{p\left( 1-p_{t}\right) }\nonumber \\
& & -\left( 1-p_{t}\right) ^{2}\Phi ^{-1}\left(
\frac{Q-f_{1t}p_{1t}^{2}-\frac{Qp_{t+1}-f_{1t}p_{t}^{2}}{p_{t}\left( 1-p_{t}\right)
}}{\left( 1-p_{t}\right) ^{2}}\right) \frac{-p_{t}^{2}\left( 1-\frac{1}{p_{t}\left(
1-p_{t}\right) }\right) }{\left( 1-p_{t}\right) ^{2}}\nonumber \\
& = & -p_{t}^{2}\Phi ^{-1}\left( f_{1t}\right) +2\frac{p_{t}^{3}}{1-p_{t}}\Phi
^{-1}\left( \frac{Qp_{t+1}-f_{1t}p_{t}^{2}}{p_{t}\left( 1-p_{t}\right) }\right)
\nonumber \\
& & +p_{t}^{2}\left( 1-\frac{1}{p_{t}\left( 1-p_{t}\right) }\right) \Phi ^{-1}\left(
\frac{Q-f_{1t}p_{1t}^{2}-\frac{Qp_{t+1}-f_{1t}p_{t}^{2}}{p_{t}\left( 1-p_{t}\right)
}}{\left( 1-p_{t}\right) ^{2}}\right)
\end{eqnarray}
\end_inset
When
\begin_inset Formula \( y \)
\end_inset
is minimized, the increase in
\begin_inset Formula \( \bar{A}_{t+1} \)
\end_inset
is maximal.
\layout Standard
Note that while the derivative of
\begin_inset Formula \( y \)
\end_inset
has been calculated, this will not always be possible.
however, the optimization routines in standard libraries are both faster
and more efficient when this derivative is available.
\layout Standard
Also note that the conversion yields a different starting point than the
actual mass selection solution.
While the frequencies are those reached by mass selection, the
\begin_inset Formula \( f_{t} \)
\end_inset
are actually different, as they are selected to maximize
\begin_inset Formula \( \bar{A}_{t+1} \)
\end_inset
in the subsequent generation.
As such, the starting point is actually a superior solution to the mass
selection solution.
\layout Subsection
Solving the Model
\layout Standard
Program
\family typewriter
dp2
\family default
is the initial solution to the problem.
At this stage, the goal is a clean and easily understood routine, and no
attempts have been made at optimization of the algorithm.
\layout Standard
The mass selection solution is taken as the starting value.
An array of bubbles is created with indices from
\begin_inset Formula \( 0 \)
\end_inset
to the total number of generations considered,
\family typewriter
totGenerations
\family default
.
Each of these bubbles is centered around the corresponding mass selection
gene frequency.
Bubble
\begin_inset Formula \( 0 \)
\end_inset
has a single bubblette, as the starting frequency is a parameter of the
problem.
Each of the other bubbles receive bubblettes to reach from
\begin_inset Formula \( .05 \)
\end_inset
beneath the mass selection value to as close as possible to
\begin_inset Formula \( 1.0 \)
\end_inset
without exceeding it, and a grain of
\begin_inset Formula \( .05 \)
\end_inset
is used for all generations.
This is far larger than is needed, but by extending beyond all conceivable
values, issues related to stepping past boundaries are avoided for the
moment.
\layout Standard
The function
\family typewriter
bestVal()
\family default
is defined, which returns the best possible increase in the value of the
objective function that can be reached from a specified bubble.
This is the change in value from that particular choice of frequency at
that generation,
\emph on
including the value from all subsequent choices.
\emph default
\layout Standard
The function works by sequentially considering a subset of the possible
choices for the
\emph on
next
\emph default
period's major gene frequency.
However, before it does so, it checks to see if this bubblette has already
been considered.
If so, it returns the previously calculated value.
Failing this, an initial guess, which will for the moment and for illustrative
purposes be assumed to be
\begin_inset Formula \( 0 \)
\end_inset
, as to the offset from the center of the next bubble is made.
The corresponding bubblette in the next generation is queried for its value,
to which the gain
\begin_inset Formula \( \Delta \bar{A}_{t+1} \)
\end_inset
resulting from this tentative choice is added, and the result placed in
\family typewriter
testVal
\family default
.
\family typewriter
testVal
\family default
is also copied to the local variable
\family typewriter
thisBest
\family default
, which stores the best value of the best step found to date.
\layout Standard
The routine then tries an offset one greater than the first considered,
and calculates its value in the same way.
If the result is better than that stored in
\family typewriter
thisBest
\family default
,
\family typewriter
thisBest
\family default
is updated, and the process repeated for the next larger offset.
\layout Standard
If the result of the first step is inferior to
\family typewriter
thisBest
\family default
, the direction is switched; the offset one less than the stating value
is considered.
\layout Standard
The process is repeated until the next value considered declines.
At this time, the prior step is recognized as the best, the step stored
in
\family typewriter
nextp
\family default
, and the value returned.
\layout Standard
It is important to note at this time that the bubblette with the highest
value will not necessarily be chosen.
\begin_float footnote
\layout Standard
In fact, it seems that this is rarely what happens.
\end_float
It is not only the value of the bubblette itself that matters, but also
the value
\emph on
added
\emph default
in stepping to that bubblette.
Thus adding
\begin_inset Formula \( .5 \)
\end_inset
while reaching a bubblette with value
\begin_inset Formula \( .3 \)
\end_inset
is more valuable than adding
\begin_inset Formula \( .2 \)
\end_inset
while reaching a bubblette with value
\begin_inset Formula \( .4 \)
\end_inset
.
Additionally, though the value of the bubblette is fixed regardless of
the step taken to reach it, the value added reaching it will vary depending
upon the prior state.
Thus different bubblettes will be chosen from different prior states.
In non-trivial problems, it is expected that there will be a tradeoff between
the stepping gain and the value of the bubblette; if this were not the
case, both could be maximized.
\layout Standard
\family typewriter
bestVal()
\family default
does have a
\begin_inset Quotes eld
\end_inset
special
\begin_inset Quotes erd
\end_inset
case for the final period.
The last choice is made in the penultimate generation.
\family typewriter
bestVal()
\family default
is called again, but recognizes that it is called for the final generations,
and simply returns
\begin_inset Formula \( \left( 2p-1\right) a \)
\end_inset
, the contribution of the major gene in the final period.
\layout Standard
The function operates recursively.
As such, querying the sole bubblette in generation
\begin_inset Formula \( 0 \)
\end_inset
for its value results in calculation of needed values for all subsequent
generations.
\layout Standard
When a solution is reached, the bubbles are
\begin_inset Quotes eld
\end_inset
collapsed.
\begin_inset Quotes erd
\end_inset
New bubbles are created centered about the path chosen, and with the grain
reduced.
Additionally, hints are stored regarding the
\begin_inset Quotes eld
\end_inset
expected
\begin_inset Quotes erd
\end_inset
path from bubblette to bubblette.
For the center bubblettes, this is simply the next centered bubblette.
For other bubblettes
\emph on
in the region calculated in the prior iteration,
\emph default
the hint is the bubblette at the same point in space which was previously
reached.
Due to the collapsing, there will be somewhat arbitrary choices made as
to which bubblettes correspond to which.
Rather then spend effort and computation on the matter, this is simply
left to the default rounding performed by Fortran: at most, it will be
off by a single state in any dimension, and adjacent states will be checked
in any event.
\layout Standard
A mininum of two bubblettes above and below the center will be created within
each bubble.
However, a check is made to determine how many states were actually checked
in that direction for that bubble, and if larger than two, this quantity
is used instead.
Furthermore, a check is made to insure that no boundary states were chosen
in the final solution--this could indicate that a better state existed
after the boundary.
In this case, the boundary is doubled, and the iteration repeated with
the same grain.
\begin_inset LatexCommand \label{boundarydoubling}
\end_inset
\begin_float margin
\layout Standard
is the repetition really needed? discuss?
\end_float
\layout Standard
The process is repeated until the grain of the probability space is satisfactori
ly small, with the final iteration yielding the reported solution.
\layout Standard
Using the equations from
\begin_inset LatexCommand \cite{Dekkers}
\end_inset
, it is possible to use standard iterative methods to find the optimal values.
However, this does not solve the problem, but rather uses iteration to
reach an already found solution.
Dekkers' method yields the following value for fifteen generations,
\layout Standard
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Without using any of the optimal control equations, dp2 yields essentally
the same results:q
\layout Subsection
Mechanics of Conversion to Frequency Space
\layout Standard
As discussed in
\begin_inset LatexCommand \ref{sec:replacechoicevars}
\end_inset
, a fundamental change to the problem has been made while converting to
frequency space, and the reduction from a two-dimensional search space
to a single-dimensional space.
While the same answer will be reached at the optimal solution, the starting
point is not the same as the mass selection solution.
\layout Paragraph
choosing f1 & f2
\layout Standard
With
\begin_inset Formula \( p_{t+1} \)
\end_inset
chosen,
\begin_inset Formula \( f_{t} \)
\end_inset
should be chosen to maximize
\begin_inset Formula \( \Delta \bar{A}_{t+1} \)
\end_inset
.
The optimization problem is
\begin_inset Formula
\begin{eqnarray}
max_{f_{t}}L & = & \Delta \bar{A}_{t+1}\nonumber \\
& & +\lambda \left[ f_{1t}p_{t}^{2}+f_{2t}p_{t}\left( 1-p_{t}\right)
-Qp_{t+1}\right] +\mu \left[ f_{1t}p_{t}^{2}+f_{2t}2p_{t}\left( 1-p_{t}\right)
+f_{3t}\left( 1-p_{t}^{2}\right) \right] \nonumber \\
& = & \frac{\sigma }{Q}\left\{ p_{t}^{2}z_{1t}+2p_{t}\left( 1-p_{t}\right)
z_{2t}+\left( 1-p_{t}^{2}z_{3t}\right) \right\} \nonumber \\
& & +\lambda \left[ f_{1t}p_{t}^{2}+f_{2t}p_{t}\left( 1-p_{t}\right)
-Qp_{t+1}\right] +\mu \left[ f_{1t}p_{t}^{2}+f_{2t}2p_{t}\left( 1-p_{t}\right)
+f_{3t}\left( 1-p_{t}^{2}\right) \right]
\end{eqnarray}
\end_inset
which would require an iterative solution, due to the presence of inverses
of the normal cumulative distribution function.
As such, it is simpler to use a routine from a standard library.
Still, such routines require boundaries for the search.
\layout Standard
\begin_inset Formula \( f_{1t} \)
\end_inset
will be used as the choice variable, with
\begin_inset Formula \( f_{2t} \)
\end_inset
determined by this choice.
In any
\begin_inset Quotes eld
\end_inset
sane
\begin_inset Quotes erd
\end_inset
choice,
\begin_inset Formula
\begin{equation}
f_{1t}>f_{2t}
\end{equation}
\end_inset
This will generally be used as the lower bound, and can be calculated as
\begin_inset Formula
\begin{equation}
f^{0}_{1tmin}=Q\frac{p_{t+1}}{p_{t}}
\end{equation}
\end_inset
\layout Standard
In the special case where setting
\begin_inset Formula \( f_{1t} \)
\end_inset
and
\begin_inset Formula \( f_{2t} \)
\end_inset
equal causes these portions to exceed
\begin_inset Formula \( Q \)
\end_inset
,
\begin_inset Formula \( f_{1t} \)
\end_inset
must be increased; the floor has been understated.
The new floor will still allocate nothing, or as close to nothing as allocated
by the algorithm, to
\begin_inset Formula \( f_{3t} \)
\end_inset
.
As such, [
\begin_inset LatexCommand \ref{dp2qcons}
\end_inset
] reduces to
\begin_inset Formula
\begin{equation}
f_{1t}p_{t}^{2}+f_{2t}2p_{t}\left( 1-p_{t}\right) =Q
\end{equation}
\end_inset
which combined with [
\begin_inset LatexCommand \ref{dp2prule}
\end_inset
] yields two equations in two unknowns for the minimal value of
\begin_inset Formula \( f_{1t} \)
\end_inset
.
As
\begin_inset Formula \( f_{1t} \)
\end_inset
rises from this value,
\begin_inset Formula \( f_{2t} \)
\end_inset
must diminish, and
\begin_inset Formula \( f_{3t} \)
\end_inset
will rise.
Solving the equations,
\begin_inset Formula
\begin{eqnarray}
\left[ \begin{array}{cc}
p_{t}^{2} & 2p_{t}\left( 1-p_{t}\right) \\
p_{t}^{2} & p_{t}\left( 1-p_{t}\right)
\end{array}\right] \left[ \begin{array}{c}
f_{1t}\\
f_{2t}
\end{array}\right] & = & \left[ \begin{array}{c}
Q\\
Qp_{t+1}
\end{array}\right] \nonumber \\
\left[ \begin{array}{cc}
p_{t} & 2\left( 1-p_{t}\right) \\
p_{t} & \left( 1-p_{t}\right)
\end{array}\right] \left[ \begin{array}{c}
f_{1t}\\
f_{2t}
\end{array}\right] & = & \frac{Q}{p_{t}}\left[ \begin{array}{c}
1\\
p_{t+1}
\end{array}\right] \nonumber \\
\left[ \begin{array}{c}
f_{1t}\\
f_{2t}
\end{array}\right] & = & \frac{Q}{p_{t}}\frac{\left[ \begin{array}{cc}
\left( 1-p_{t}\right) & -2\left( 1-p_{t}\right) \\
-p_{t} & p_{t}
\end{array}\right] }{p_{t}\left( 1-p_{t}\right) -p_{t}2\left( 1-p_{t}\right) }\left[
\begin{array}{c}
1\\
p_{t+1}
\end{array}\right] \nonumber \\
& = & -\frac{Q}{p_{t}}\frac{\left[ \begin{array}{cc}
\left( 1-p_{t}\right) & -2\left( 1-p_{t}\right) \\
-p_{t} & p_{t}
\end{array}\right] }{p_{t}\left( 1-p_{t}\right) }\left[ \begin{array}{c}
1\\
p_{t+1}
\end{array}\right] \nonumber \\
& = & -\frac{Q}{p_{t}^{2}\left( 1-p_{t}\right) }\left[ \begin{array}{c}
\left( 1-p_{t}\right) -2\left( 1-p_{t}\right) p_{t+1}\\
p_{t}+p_{t}p_{t+1}
\end{array}\right] \nonumber \\
& = & -\frac{Q}{p_{t}^{2}\left( 1-p_{t}\right) }\left[ \begin{array}{c}
\left( 1-p_{t}\right) \left( 1-2p_{t+1}\right) \\
p_{t}\left( 1+p_{t+1}\right)
\end{array}\right] \nonumber \\
& = & \left[ \begin{array}{c}
\frac{Q\left( 2p_{t+1}-1\right) }{p_{t}^{2}}\\
\frac{Q\left( 1+p_{t+1}\right) }{p_{t}\left( 1-p_{t}\right) }
\end{array}\right]
\end{eqnarray}
\end_inset
of which only the first is used.
\layout Standard
An upper bound must similarly be found.
For
\begin_inset Formula
\begin{equation}
p_{t}<\sqrt{Q}
\end{equation}
\end_inset
it is possible to use all of the homozygotes, and
\begin_inset Formula
\begin{equation}
f_{1tmax}^{0}=1
\end{equation}
\end_inset
For larger values of
\begin_inset Formula \( p_{t} \)
\end_inset
,
\begin_inset Formula
\begin{equation}
f_{1tmax}^{1}=\frac{Q}{p_{t}^{2}}
\end{equation}
\end_inset
selects entirely from this type, and is used as the maximum.
\layout Standard
For this finite model, each generation has its own bubble; the next bubble
is known with certainty.
For the initial pass, sufficient bubblettes are used such that the full
range
\begin_inset Formula \( \left( 0,1\right) \)
\end_inset
is available in each choice generation, save that a single bubblette is
available for generation
\begin_inset Formula \( 0 \)
\end_inset
, as its gene frequency is predetermined as part of the problem.
\layout Standard
An initial grain of
\begin_inset Formula \( .05 \)
\end_inset
is set for
\begin_inset Formula \( p_{t} \)
\end_inset
in all generations, and the bubblette of generation
\begin_inset Formula \( 0 \)
\end_inset
is asked for its value.
\layout Standard
When a bubblette is asked for its value, it firsts checks to see if it has
been calculated already.
If so, it returns its value.
If not, it checks itself for a hint from prior generations, choosing a
hint of
\begin_inset Formula \( 0 \)
\end_inset
if none is found.
Having established the hint, the a
\begin_inset Quotes eld
\end_inset
sanity check
\begin_inset Quotes erd
\end_inset
\begin_float margin
\layout Standard
discuss danger of check & missing optima
\end_float
is made on the hint: the gene frequency will never decrease from generation
to generation; any such step is wrong.
As such, if the hint suggests reducing the next generation's frequency
below the present level, it is rejected, and increase to a sane level.
Similarly, a sanity check is made to insure that the
\begin_inset Formula \( p_{t+1} \)
\end_inset
considered is actually a possible transition.
\layout Standard
This achieved, the value of the hint is calculated.
The bubblette from the next generation corresponding to the hint is queried,
and
\begin_inset Formula \( \Delta \bar{A}_{t+1} \)
\end_inset
is added to this value.
This is stored as the best tentative value, and the next higher bubblette
in the next generation is checked.
If that bubblette is better, it becomes the new hint, and the search proceeds
in that direction as long as progress is made.
If inferior, the next lower bubblette is checked, and the search proceeds
in that direction as long as it is successful.
However, if the initial hint is less than
\begin_inset Formula \( 0 \)
\end_inset
, either from a sanity check or the initial hint, the search starts first
in the negative direction, and switches to positive if appropriate.
\layout Standard
Once a bubblette selects a value in this manner, it stores the final hint,
indicating the next state chosen, its value, and the fact of calculation.
It then returns.
\layout Standard
After the initial bubblette returns a value, the optimal path can be found
by stepping from bubble to bubble, which is the method used by
\family typewriter
candidateFromBubbles
\family default
to form a reportable solution.
\layout Standard
With a solution found, another iteration is made.
If any of the steps chosen as optimal were on the boundary of a bubblette,
it is not clear that a further step was not desirable.
That boundary is doubled, and the process repeated with the same grain.
\layout Standard
If no boundaries were used, the optimal solution becomes the base frequency
of the next solution.
\layout Standard
Initially, the boundaries were set by doubling one more than the highest
state actually used in that direction (above or below the base), with a
minimum of ten.
This resulted in a solution time of
\begin_inset Formula \( 40.1 \)
\end_inset
seconds (also using zero rather than the actual value for hints).
Changing this to one larger than was actually used, with a minimum of two,
significantly improved performance, to
\begin_inset Formula \( 27.7 \)
\end_inset
seconds, or a reduction in processing of about one third.
Enabling the hints further reduced processing time to
\begin_inset Formula \( 16.7 \)
\end_inset
seconds, only a third of the initial amount.
Interestingly, storing
\begin_inset Formula \( f_{1t} \)
\end_inset
to use as a hint in subsequent iterations actually
\emph on
increased
\emph default
the execution time by a marginal amount, to
\begin_inset Formula \( 16.9 \)
\end_inset
seconds.
\layout Subsection
Adding Discounting
\layout Standard
To this point, the only concern has been the maximum amount of
\emph on
progress
\emph default
that can be made, and only the final generation has been considered.
It is not difficult, however, to modify the work done to this point to
allow for an infinite horizon.
\layout Standard
Previously, the program considered the effect of the major gene only in
the final generation, and added the gain in polygenic value from each generatio
n, creating a sum equal to the value in the final generation.
\layout Standard
It is a property of the genetic model that changes in polygenic value are
permanent.
As such, a change in generation
\begin_inset Formula \( t \)
\end_inset
increases all generations by the same amount.
The discounted value of an increase
\begin_inset Formula \( b \)
\end_inset
, with a discounted value
\begin_inset Formula
\begin{equation}
d=1-r
\end{equation}
\end_inset
is not difficult to calculate for a finite horizon:
\layout Standard
\begin_inset Formula
\begin{eqnarray}
\sum _{i=0}^{T}bd^{i} & = & \sum _{i=0}^{\infty }bd^{i}-\sum _{i=T+1}^{\infty
}bd^{i}\nonumber \\
& = & \sum _{i=0}^{\infty }bd^{i}-d^{T+1}\sum _{i=0}^{\infty }bd^{i}\nonumber \\
& = & b\frac{1-d^{T+1}}{1-d}\label{identfinitediscount}
\end{eqnarray}
\end_inset
\layout Standard
To calculate a discounted value, then,
\family typewriter
bestVal()
\family default
need only be modified to apply the identity in [
\begin_inset LatexCommand \ref{identfinitediscount}
\end_inset
] to the calculated value for
\begin_inset Formula \( \Delta \bar{A}_{t+1} \)
\end_inset
, discount this value and that returned from
\family typewriter
nextVal()
\family default
, and add the current value of the major gene.This is accomplished by a simple
if/then structure in
\family typewriter
bestVal()
\family default
.
To stay with a single code base, these actions are taken only if the variable
discount has a non-zero value.
The only other change required is to change the print routine for tentative
solutions such that the present value of each generation and its future
is displayed.
The present value
\emph on
of the choice made
\emph default
is reported for each generation.
This is not the same as the present value of the generation; the value
of the current state is not included.
By calculating in this manner, it is easier to compare the relative value
of choices later when choosing whether or not to test.
\layout Subsection
An Infinite Horizon
\layout Standard
Extending the problem to an infinite horizon is also straightforward, and
can be done in a number of different ways.
\latex latex
\backslash
sout{Heading: simplistic approach}
\layout Standard
The simplest approach, which will be used for the moment, is simply to use
the solution for the
\begin_inset Formula \( n \)
\end_inset
-generation problem as a starting point for the
\begin_inset Formula \( n+1 \)
\end_inset
-generation solution.
The starting value for final
\begin_inset Formula \( p_{t} \)
\end_inset
is taken as increasing by half as much as in the prior generation, but
no more than half of the distance to
\begin_inset Formula \( 1.0 \)
\end_inset
.
\layout Standard
Due to steep discounting, future generations eventually have very little
present value.
Furthermore, the
\emph on
difference
\emph default
between the present value of subsequent far off generations declines.
As such, for any arbitrarily small
\begin_inset Formula \( \epsilon \)
\end_inset
, there exists an
\begin_inset Formula \( n \)
\end_inset
such that the difference between the present values of proceeding for
\begin_inset Formula \( n \)
\end_inset
and for
\begin_inset Formula \( n+1 \)
\end_inset
generations is less than
\begin_inset Formula \( \epsilon \)
\end_inset
.
This is the first approach taken, and requires minimal modifications such
that the prior version of the program is placed in a loop, which exits
when the gain between subsequent generations is less than the specified
convergence criterion.
\layout Subsubsection
Using Mass Selection
\layout Standard
While the simplistic approach could work, it is not guaranteed to without
significant work, as the routines used have an upper bound on the
\begin_inset Formula \( p_{t} \)
\end_inset
for which inverses of distribution functions may be calculated.
However, a better method exists.
In the simple approach, the entire value of the added generation is an
increase, though it is possibly offset by different actions in prior generation
s (the steps that are optimal for
\begin_inset Formula \( n \)
\end_inset
and
\begin_inset Formula \( n+1 \)
\end_inset
generations are not the same, and thus the first
\begin_inset Formula \( n \)
\end_inset
steps of the
\begin_inset Formula \( n+1 \)
\end_inset
generation solution are worth less than the
\begin_inset Formula \( n \)
\end_inset
generation solution).
A more efficient solution is to instead of ignoring generations after
\begin_inset Formula \( n \)
\end_inset
to switch to mass selection, and calculate the present value in that manner.
This simplifies the calculations in [
\begin_inset LatexCommand \ref{identfinitediscount}
\end_inset
], which becomes
\begin_inset Formula
\begin{equation}
\label{identinfdiscount}
\sum _{i=0}^{\infty }bd^{i}=\frac{b}{1-d}
\end{equation}
\end_inset
\layout Standard
The only further modification required is to change
\family typewriter
bestVal()
\family default
such that in the final generation, it returns the present discounted value
of future generations under mass selection.
\layout Paragraph
The Value of Mass Selection
\layout Standard
Starting with
\begin_inset Formula
\begin{equation}
\left[ \begin{array}{ccccc}
x_{1t} & -x_{2t} & & & -\frac{a}{\sigma }\\
x_{1t} & & -x_{3t} & & -2\frac{a}{\sigma }\\
& & & \Phi \left( -x_{1t}\right) p_{t}^{2}+\Phi \left( -x_{2t}\right) p_{t}\left(
1-p_{t}\right) +\Phi \left( -x_{3t}\right) \left( 1-p_{t}\right) ^{2} & -Q
\end{array}\right] =0
\end{equation}
\end_inset
yields
\begin_inset Formula
\begin{equation}
\Phi \left( -x_{1t}\right) p_{t}^{2}+\Phi \left( -x_{1t}+\frac{a}{\sigma }\right)
p_{t}\left( 1-p_{t}\right) +\Phi \left( -x_{1t}+2\frac{a}{\sigma }\right) \left(
1-p_{t}\right) ^{2}-Q
\end{equation}
\end_inset
or
\begin_inset Formula
\[
L=\Phi \left( -x_{1t}\right) p_{t}^{2}+\Phi \left( -x_{2t}\right) p_{t}\left(
1-p_{t}\right) +\Phi \left( -x_{3t}\right) \left( 1-p_{t}\right) ^{2}-Q\]
\end_inset
and
\begin_inset Formula
\begin{eqnarray}
L' & = & x_{1t}\phi ^{2}\left( -x_{1t}\right) p_{t}^{2}+\left( x_{1t}-\frac{a}{\sigma
}\right) \phi ^{2}\left( -x_{1t}+\frac{a}{\sigma }\right) p_{t}\left( 1-p_{t}\right)
\nonumber \\
& & +\left( x_{1t}-2\frac{a}{\sigma }\right) \phi ^{2}\left(
-x_{1t}+2\frac{a}{\sigma }\right) \left( 1-p_{t}\right) ^{2}
\end{eqnarray}
\end_inset
These equations are fed to a numeric routine for solution; reducing to
the single variable
\begin_inset Formula \( x_{1t} \)
\end_inset
speeds calculation.
The resultant values are calculated and discounted until
\begin_inset Formula \( p_{t+1} \)
\end_inset
is within the constant
\family typewriter
peps
\family default
, the limit of resolution for gene frequency, of
\family typewriter
\begin_inset Formula \( 1.0 \)
\end_inset
.
\begin_inset Formula \( p_{t} \)
\end_inset
\family default
is then accepted as being equal to one, and the gain in polygenic value
becomes fixed for all later generations, and [
\begin_inset LatexCommand \ref{identinfdiscount}
\end_inset
] is used to add the value of all future gain.
\begin_float margin
\layout Standard
The
\family typewriter
pcrit
\family default
limit doesn't make sense in the finite horizon, as state isn't Markov
\end_float
\layout Subsection
Testing Costs
\layout Standard
The results so far consider only the revenue from the breeding program,
and not the costs.
Realistically, a cost should be imposed when animals are tested, and the
breeding program should only continue when the benefits exceed the cost,
the benefit being the gain
\emph on
in excess of
\emph default
the gain from mass selection.
The algorithm changes only slightly to handle this variation: the best
possible breeding choice is still found, but its value is compared to the
value of switching to mass selection.
If it does not beat mass selection by the testing cost
\begin_inset Formula \( c \)
\end_inset
, the switch is made.
Using a discount rate of
\begin_inset Formula \( 8\% \)
\end_inset
and a test cost of
\begin_inset Formula \( .1 \)
\end_inset
with Dekkers' parameters, the transition to mass selection occurs after
generation
\begin_inset Formula \( 7 \)
\end_inset
.
\layout Subsection
Changing the Horizon
\layout Standard
The largest computational cost is not in calculating the values of the states,
but in preparing the bubblettes for this computation.
Once it is known that states beyond a given generation are not used, there
is no reason to continue calculating these states.
Similarly, if a breeding program has not switched to mass selection, a
longer program may be desirable.
The control variable
\family typewriter
smartShrinkGens
\family default
is added to handle this situation.
\layout Standard
With
\family typewriter
smartShrinkGens
\family default
set, if mass selection is not chosen in the final choice generation, the
time horizon is increased by one.
The solution of the current iteration, augmented by mass selection for
the final generation, is taken as the center, and the next iteration is
run with the same grain.
\layout Standard
Conversely, if the switch to mass selection occurs before the final generation,
the generation in which the switch occurs becomes the final generation.
However, the grain is reduced, as the available states are a subset of
the states already considered.
\layout Standard
This final model can be expressed as
\begin_inset Formula
\begin{eqnarray}
\max _{T,\left\{ f_{t}:0\leq <T-1\right\} }L & = & \sum _{t=0}^{T-1}\left( 1-d\right)
^{t+1}\left[ a\left( 2p_{t+1}\left( f_{t}\right) -1\right) +\frac{\Delta
\bar{A}_{t+1}\left( f_{t}\right) }{d}-c\right] \nonumber \\
& & +\left( 1-d\right) ^{T}\left[ \frac{a}{d}+\frac{\Delta \bar{A}}{d^{2}}\right]
\label{dp2finalobj}
\end{eqnarray}
\end_inset
\begin_float margin
\layout Standard
discuss 20% penalty for
\begin_inset Formula \( \pm 3 \)
\end_inset
rather than
\begin_inset Formula \( \pm 2 \)
\end_inset
\end_float
\layout Standard
\latex latex
\backslash
pagebreak
\layout Section
Two Dimensions: Adding Disequilibrium
\layout Standard
The model given in [
\begin_inset LatexCommand \ref{dp2finalobj}
\end_inset
] is incomplete in several ways.
The first to be considered is that of
\emph on
gametic phase disequilibrium
\emph default
\latex latex
\backslash
jd{
\latex default
between the major gene and polygenes
\latex latex
}
\latex default
.
\emph on
\begin_inset LatexCommand \index{gametic phase disequilibrium}
\end_inset
\emph default
While not considered to this point, the selection intensity on the polygenes
is weaker for type BB than for Bb, which is in turn weaker than for bb.
This is a simple consequence of the higher fractions selected from BB and
Bb; animals with a lower polygenic value than in bb survive selection.
\layout Standard
As with the first model, the actual choice variables,
\begin_inset Formula \( f_{t} \)
\end_inset
, will not be used.
Rather than a single state variable and an internal optimization given
that variable, a pair of state variables will now be used:
\begin_inset Formula \( p_{t} \)
\end_inset
, and the difference between the
\latex latex
\backslash
jd{
\latex default
average
\latex latex
}
\latex default
polygenic values
\latex latex
\backslash
jd{
\latex default
associated with the B and b gamets,
\latex latex
}
\latex default
\begin_inset Formula \( \bar{A}_{B,t}-\bar{A}_{b,t} \)
\end_inset
.
\layout Subsection
Gametic Phase Disequilibrium
\layout Standard
The
\begin_float margin
\layout Standard
should this go in an earlier section?
\end_float
single dimensional approache ignores the effects of
\begin_inset Quotes eld
\end_inset
gametic phase disequilibrium,
\begin_inset Quotes erd
\end_inset
\begin_inset LatexCommand \index{disequilibrium, gametic phase}
\end_inset
, a well known consequence of selection.
\begin_inset LatexCommand \cite{IQG}
\end_inset
.
Under selection, the superior homozygotes, BB, are subject to a lesser
selection intensity for polygenic effects than the other types,
\begin_inset LatexCommand \cite{Dekkers}
\end_inset
and as a consequence, have a lower polygenic value.
Rather than a single
\latex latex
\backslash
jd{
\latex default
average
\latex latex
\latex default
polygenic
\latex latex
}
\latex default
value
\begin_inset Formula \( \bar{A}_{t} \)
\end_inset
,
\latex latex
\backslash
so{
\latex default
there are
\latex latex
}
\latex default
now two values,
\begin_inset Formula \( \bar{A}_{B,t} \)
\end_inset
and
\begin_inset Formula \( \bar{A}_{b,t} \)
\end_inset
\latex latex
\backslash
jd{
\latex default
must be distiguished
\latex latex
}
\latex default
, reflecting
\latex latex
\backslash
jd{
\latex default
average
\latex latex
\latex default
polygenic values for gametes carrying the B and b allelles, respectively.
\latex latex
}
\latex default
As each individual gets two gametes, one from each parent, the average
polygenic values for BB, Bb, and bb are then, respectively,
\begin_inset Formula \( 2\bar{A}_{B,t} \)
\end_inset
,
\begin_inset Formula \( \bar{A}_{B,t}+\bar{A}_{b,t} \)
\end_inset
, and
\begin_inset Formula \( 2\bar{A}_{b,t} \)
\end_inset
.
The overall value is then a weighted average,
\begin_inset Formula
\begin{equation}
\bar{A}_{t}=2p_{t}\bar{A}_{B,t}+2\left( 1-p_{t}\right) \bar{A}_{b,t}
\end{equation}
\end_inset
\layout Standard
Similarly
\begin_float footnote
\layout Standard
similar or similarly? does it modify
\begin_inset Quotes eld
\end_inset
problem
\begin_inset Quotes erd
\end_inset
or
\begin_inset Quotes eld
\end_inset
written
\begin_inset Quotes erd
\end_inset
?
\end_float
to the single dimensional problem, these average values can be written
as recursive equations:
\begin_inset Formula
\begin{equation}
\label{diseqABt1}
\bar{A}_{B,t+1}=\frac{f_{1t}p_{t}^{2}\left( \bar{A}_{B,t}+\frac{1}{2}i_{1t}\sigma
\right) +f_{2t}p_{t}\left( 1-p_{t}\right) \frac{1}{2}\left(
\bar{A}_{B,t}+\bar{A}_{b,t}+i_{2t}\sigma \right) }{f_{1t}p_{t}^{2}+f_{2t}p_{t}\left(
1-p_{t}\right) }
\end{equation}
\end_inset
and
\begin_inset Formula
\begin{equation}
\label{diseqAbt1}
\bar{A}_{b,t+1}=\frac{f_{2t}p_{t}\left( 1-p_{t}\right) \frac{1}{2}\left(
\bar{A}_{B,t}+\bar{A}_{b,t}+i_{2t}\sigma \right) +f_{3t}\left( 1-p_{t}\right)
^{2}\left( \bar{A}_{b,t}+\frac{1}{2}i_{3t}\sigma \right)
}{1-f_{1t}p_{t}^{2}-f_{2t}p_{t}\left( 1-p_{t}\right) }
\end{equation}
\end_inset
\begin_float margin
\layout Standard
jack: is
\begin_inset Formula \( \bar{A}_{b,t+1} \)
\end_inset
correct?
\end_float
the first of which is Dekkers' equation 10.
Note that this usage differs from Dekkers' introduction of
\begin_inset Formula \( W_{B,t}=p_{t}\bar{A}_{B,t} \)
\end_inset
and
\begin_inset Formula \( W_{b,t}=\left( 1-p_{t}\right) \bar{A}_{b,t} \)
\end_inset
.
While this change was advantageous for the use of optimal control, it would
introduce complications for the methods below.
Particularly, the raw polygenic values have the same units, which allows
their difference to be defined, while the
\begin_inset Formula \( W \)
\end_inset
are weighted to reflect their contributions to the overal average polygenic
value.
\layout Subsection
Finding the State Variables
\layout Standard
As before, there are more choice variables than state variables, and non-lineari
ty makes the actual choice variables impractical for dynamic programming.
Two
\latex latex
\backslash
sout{
\latex default
choice
\latex latex
}
\backslash
rh{
\latex default
state
\latex latex
}
\latex default
variables are now needed, and the mechanism for maximising [
\begin_inset LatexCommand \ref{dpsiminfanextoff1}
\end_inset
] is now irrelevant.
The choice of
\begin_inset Formula \( p_{t+1} \)
\end_inset
and
\latex latex
\backslash
rh{
\latex default
a single additional variable for period
\begin_inset Formula \( t+1 \)
\end_inset
\latex latex
}
\latex default
d
\latex latex
\backslash
so{
\latex default
such other state variable as is chosen
\latex latex
}
\latex default
will
\emph on
exactly
\emph default
define
\begin_inset Formula \( f_{mt} \)
\end_inset
,
\latex latex
\backslash
jd{
\latex default
as [
\begin_inset LatexCommand \ref{siminfqcns}
\end_inset
], [
\begin_inset LatexCommand \ref{diseqABt1}
\end_inset
], and [
\begin_inset LatexCommand \ref{diseqAbt1}
\end_inset
] create three equations in the three unknowns
\begin_inset Formula \( f_{mt} \)
\end_inset
\latex latex
}
\latex default
.
\layout Standard
The new fitness function for a given generation, comparable to [
\begin_inset LatexCommand \ref{siminfobjfn}
\end_inset
] is now
\begin_inset Formula
\begin{equation}
\label{diseqrawobjfn}
\bar{G}_{t}=a\left( 2p_{t}-1\right) +2p_{t}\bar{A}_{B,t}+2\left( 1-p_{t}\right)
\bar{A}_{b,t}
\end{equation}
\end_inset
However, subsituting [
\begin_inset LatexCommand \ref{diseqABt1}
\end_inset
] and [
\begin_inset LatexCommand \ref{diseqAbt1}
\end_inset
] into[
\begin_inset LatexCommand \ref{diseqrawobjfn}
\end_inset
] does not yield a simple result such as [
\begin_inset LatexCommand \ref{siminfarule}
\end_inset
], in which the change
\latex latex
\backslash
jd{
\latex default
in
\begin_inset Formula \( \bar{A} \)
\end_inset
\latex latex
}
\latex default
is easily isolated.
Instead, a dependence upon two variables of the prior generation
\latex latex
\backslash
jd{
\latex default
,
\begin_inset Formula \( \bar{A}_{B,t} \)
\end_inset
and
\begin_inset Formula \( \bar{A}_{b,t} \)
\end_inset
,
\latex latex
}
\latex default
remains.
In order to create a useful algorithm, it is necessary to completely isolate
the effects of the past
\latex latex
\backslash
jd{
\latex default
states
\latex latex
}
\latex default
and the
\latex latex
\backslash
so{
\latex default
changes made
\latex latex
}
\latex default
\latex latex
\backslash
jd{
\latex default
choices made; it is necessary to have an expression such as [**] which describes
the effect of the choice on the objective function
\latex latex
}
\latex default
.
\layout Standard
One way of doing this is to find an expression for
\begin_inset Formula \( \bar{A}_{B,t+1}-\bar{A}_{b,t+1} \)
\end_inset
\latex latex
\backslash
rh{
\latex default
in terms of
\begin_inset Formula \( \bar{A}_{B,t}-\bar{A}_{b,t} \)
\end_inset
,
\begin_inset Formula \( p_{t} \)
\end_inset
,
\begin_inset Formula \( p_{t+1} \)
\end_inset
, and
\begin_inset Formula \( f_{t} \)
\end_inset
.
Using the relation that
\latex latex
}
\latex default
\begin_inset Formula
\begin{equation}
i=\frac{z}{f}
\end{equation}
\end_inset
\begin_inset LatexCommand \cite[equation 11.5]{iqg}
\end_inset
\latex latex
\backslash
rh{
\latex default
, noting that the denominator of
\begin_inset LatexCommand \ref{diseqABt1}
\end_inset
is equal to
\begin_inset Formula \( Qp_{t+1}, \)
\end_inset
and rearranging terms,
\latex latex
}
\layout Standard
\begin_inset Formula
\begin{eqnarray}
\bar{A}_{B,t+1} & = & \frac{f_{1t}p_{t}^{2}\bar{A}_{B,t}+f_{2t}p_{t}\left(
1-p_{t}\right) \frac{1}{2}\left( \bar{A}_{B,t}+\bar{A}_{b,t}\right)
+p_{t}^{2}\frac{1}{2}z_{1t}\sigma +p_{t}\left( 1-p_{t}\right) \frac{1}{2}z_{2t}\sigma
}{Qp_{t+1}}\nonumber \\
& = & \frac{\left[ f_{1t}p_{t}^{2}+f_{2}p_{t}\left( 1-p_{t}\right) \right]
\bar{A}_{B,t}-f_{2t}p_{t}\left( 1-p_{t}\right) \frac{1}{2}\left(
\bar{A}_{B,t}-\bar{A}_{b,t}\right) +p_{t}^{2}\frac{1}{2}z_{1t}\sigma +p_{t}\left(
1-p_{t}\right) \frac{1}{2}z_{2t}\sigma }{Qp_{t+1}}\nonumber \\
& = & \frac{Qp_{t+1}\bar{A}_{B,t}-f_{2t}p_{t}\left( 1-p_{t}\right)
\frac{1}{2}d_{t}+p_{t}^{2}\frac{1}{2}z_{1t}\sigma +p_{t}\left( 1-p_{t}\right)
\frac{1}{2}z_{2t}\sigma }{Qp_{t+1}}\\
& = & \bar{A}_{B,t}-\frac{f_{2t}p_{t}\left( 1-p_{t}\right)
}{2Qp_{t+1}}d_{t}+\frac{p_{t}^{2}z_{1t}+p_{t}\left( 1-p_{t}\right)
z_{2t}}{2Qp_{t+1}}\sigma
\end{eqnarray}
\end_inset
similarly,
\layout Standard
\begin_inset Formula
\begin{eqnarray}
\bar{A}_{b,t+1} & = & \frac{f_{2t}p_{t}\left( 1-p_{t}\right) \frac{1}{2}\left(
\bar{A}_{B,t}+\bar{A}_{b,t}\right) +p_{t}\left( 1-p_{t}\right) \frac{1}{2}z_{2t}\sigma
+f_{3t}\left( 1-p_{t}\right) ^{2}\bar{A}_{b,t}+\left( 1-p_{t}\right)
^{2}\frac{1}{2}z_{3t}\sigma }{Q-Qp_{t+1}}\nonumber \\
& = & \frac{f_{2t}p_{t}\left( 1-p_{t}\right) \frac{1}{2}\left(
\bar{A}_{B,t}-\bar{A}_{b,t}\right) +\left[ f_{3t}\left( 1-p_{t}\right)
^{2}+f_{2t}p_{t}\left( 1-p_{t}\right) \right] \bar{A}_{b,t}}{Q\left( 1-p_{t+1}\right)
}\nonumber \\
& & +\frac{p_{t}\left( 1-p_{t}\right) \frac{1}{2}z_{2t}+\left( 1-p_{t}\right)
^{2}\frac{1}{2}z_{3t}}{Q\left( 1-p_{t+1}\right) }\sigma \nonumber \\
& = & \frac{\left[ f_{3t}\left( 1-p_{t}\right) ^{2}+f_{2t}p_{t}\left( 1-p_{t}\right)
\right] }{Q\left( 1-p_{t+1}\right) }\bar{A}_{b,t}+\frac{1}{2}\frac{f_{2t}p_{t}\left(
1-p_{t}\right) }{Q\left( 1-p_{t+1}\right) }d_{t}\nonumber \\
& & +\frac{1}{2}\frac{p_{t}\left( 1-p_{t}\right) z_{2t}+\left( 1-p_{t}\right)
^{2}z_{3t}}{Q\left( 1-p_{t+1}\right) }\sigma \nonumber \\
& = & \frac{\left[ Q-f_{1t}p_{t}^{2}-f_{2t}2p_{t}\left( 1-p_{t}\right)
+f_{2t}p_{t}\left( 1-p_{t}\right) \right] }{Q\left( 1-p_{t+1}\right)
}\bar{A}_{b,t}+\frac{1}{2}\frac{f_{2t}p_{t}\left( 1-p_{t}\right) }{Q\left(
1-p_{t+1}\right) }d_{t}\nonumber \\
& & +\frac{1}{2}\frac{p_{t}\left( 1-p_{t}\right) z_{2t}+\left( 1-p_{t}\right)
^{2}z_{3t}}{Q\left( 1-p_{t+1}\right) }\sigma \nonumber \\
& = & \bar{A}_{b,t}+\frac{1}{2}\frac{f_{2t}p_{t}\left( 1-p_{t}\right) }{Q\left(
1-p_{t+1}\right) }d_{t}+\frac{1}{2}\frac{p_{t}\left( 1-p_{t}\right) z_{2t}+\left(
1-p_{t}\right) ^{2}z_{3t}}{Q\left( 1-p_{t+1}\right) }\sigma
\end{eqnarray}
\end_inset
yielding
\begin_inset Formula
\begin{eqnarray}
d_{t+1} & \equiv & \bar{A}_{B,t+1}-\bar{A}_{b,t+1}\nonumber \\
& = & \bar{A}_{B,t}-\frac{1}{2Q}\frac{f_{2t}p_{t}\left( 1-p_{t}\right)
}{p_{t+1}}d_{t}+\frac{1}{2Q}\frac{p_{t}^{2}z_{1t}+p_{t}\left( 1-p_{t}\right)
z_{2t}}{p_{t+1}}\sigma \nonumber \\
& & -\bar{A}_{b,t}-\frac{1}{2Q}\frac{f_{2t}p_{t}\left( 1-p_{t}\right) }{\left(
1-p_{t+1}\right) }d_{t}-\frac{1}{2Q}\frac{p_{t}\left( 1-p_{t}\right) z_{2t}+\left(
1-p_{t}\right) ^{2}z_{3t}}{\left( 1-p_{t+1}\right) }\sigma \nonumber \\
& = & d_{t}-\frac{1}{2Q}\left[ \frac{f_{2t}p_{t}\left( 1-p_{t}\right)
}{p_{t+1}}+\frac{f_{2t}p_{t}\left( 1-p_{t}\right) }{\left( 1-p_{t+1}\right) }\right]
d_{t}\nonumber \\
& & +\frac{1}{2Q}\left[ \frac{p_{t}^{2}z_{1t}+p_{t}\left( 1-p_{t}\right)
z_{2t}}{p_{t+1}}-\frac{p_{t}\left( 1-p_{t}\right) z_{2t}+\left( 1-p_{t}\right)
^{2}z_{3t}}{\left( 1-p_{t+1}\right) }\right] \sigma \nonumber \\
& = & d_{t}-\frac{1}{2Q}f_{2t}p_{t}\left( 1-p_{t}\right) \left[
\frac{1-p_{t+1}+p_{t+1}}{p_{t+1}\left( 1-p_{t+1}\right) }\right] d_{t}\nonumber \\
& & +\frac{1}{2Q}\left[ \frac{p_{t}^{2}}{p_{t+1}}z_{1t}+p_{t}\left( 1-p_{t}\right)
\left[ \frac{1}{p_{t+1}}-\frac{1}{1-p_{t+1}}\right] z_{2t}-\frac{\left( 1-p_{t}\right)
^{2}}{\left( 1-p_{t+1}\right) }z_{3t}\right] \sigma \nonumber \\
& = & d_{t}-\frac{1}{2Q}f_{2t}\frac{p_{t}\left( 1-p_{t}\right) }{p_{t+1}\left(
1-p_{t+1}\right) }d_{t}\nonumber \\
& & +\frac{1}{2Q}\left[ \frac{p_{t}^{2}}{p_{t+1}}z_{1t}+p_{t}\left( 1-p_{t}\right)
\left[ \frac{1}{p_{t+1}}-\frac{1}{1-p_{t+1}}\right] z_{2t}-\frac{\left( 1-p_{t}\right)
^{2}}{\left( 1-p_{t+1}\right) }z_{3t}\right] \sigma \label{diseqdrule}
\end{eqnarray}
\end_inset
\layout Standard
\begin_inset Formula
\begin{equation}
\label{diseqABrule}
\bar{A}_{B,t+1}-\bar{A}_{B,t}\jd {-}\frac{f_{2t}p_{t}\left( 1-p_{t}\right)
}{2Qp_{t+1}}\left( \bar{A}_{B,t}-\bar{A}_{b,t}\right)
+\frac{f_{1t}p_{t}^{2}i_{1t}\sigma +f_{2t}p_{t}\left( 1-p_{t}\right) i_{2t}\sigma
}{2Qp_{t+1}}
\end{equation}
\end_inset
\layout Standard
Before proceeding to
\begin_inset Formula \( \bar{A}_{b,t+1} \)
\end_inset
, it is useful to define
\begin_inset Formula
\begin{equation}
d_{t}\equiv \bar{A}_{B,t}-\bar{A}_{b,t}
\end{equation}
\end_inset
and derive
\begin_inset Formula
\begin{eqnarray}
1-p_{t+1} & = & 1-\frac{1}{Q}\left\{ f_{1t}p_{t}^{2}+f_{2t}p_{t}\left( 1-p_{t}\right)
\right\} \nonumber \\
& = & \frac{Q-f_{1t}p_{t}^{2}-f_{2t}p_{t}\left( 1-p_{t}\right) }{Q}\nonumber \\
& = & \frac{f_{1}p_{t}^{2}+f_{2t}2p_{t}\left( 1-p_{t}\right) +f_{3t}\left(
1-p_{t}^{2}\right) -f_{1t}p_{t}^{2}-f_{2t}p_{t}\left( 1-p_{t}\right) }{Q}\nonumber \\
& = & \frac{1}{Q}\left\{ f_{2t}p_{t}\left( 1-p_{t}\right) +f_{3t}\left(
1-p_{t}^{2}\right) \right\}
\end{eqnarray}
\end_inset
\begin_inset Formula
\begin{eqnarray}
Q\left( 1-p_{t+1}\right) \bar{A}_{b,t+1} & = & f_{2t}p_{t}\left( 1-p_{t}\right)
\frac{1}{2}\left( \bar{A}_{B,t}+\bar{A}_{b,t}+i_{2t}\sigma \right) \nonumber \\
& & +f_{3t}\left( 1-p_{t}\right) ^{2}\left( \bar{A}_{b,t}+\frac{1}{2}i_{3t}\sigma
\right) \nonumber \\
& = & \left[ f_{2t}p_{t}\left( 1-p_{t}\right) +f_{3t}\left( 1-p_{t}^{2}\right)
\right] \bar{A}_{b,t}+f_{2t}p_{t}\left( 1-p_{t}\right) \frac{1}{2}\left(
\bar{A}_{B,t}-\bar{A}_{b,t}\right) \nonumber \\
& & +f_{2t}p_{t}\left( 1-p_{t}\right) \frac{1}{2}i_{3t}\sigma +f_{3t}\left(
1-p_{t}\right) ^{2}\frac{1}{2}i_{3t}\sigma \nonumber \\
& = & Q\left( 1-p_{t+1}\right) \bar{A}_{b,t}+f_{2t}p_{t}\left( 1-p_{t}\right)
\frac{1}{2}\left( \bar{A}_{B,t}-\bar{A}_{b,t}\right) \nonumber \\
& & +f_{2t}p_{t}\left( 1-p_{t}\right) \frac{1}{2}i_{3t}\sigma +f_{3t}\left(
1-p_{t}\right) ^{2}\frac{1}{2}i_{3t}\sigma
\end{eqnarray}
\end_inset
\layout Standard
\begin_inset Formula
\begin{equation}
\label{diseqAbrule}
\bar{A}_{b,t+1}-\bar{A}_{b,t}=\frac{f_{2t}p_{t}\left( 1-p_{t}\right) }{2Q\left(
1-p_{t+1}\right) }\left( \bar{A}_{B,t}-\bar{A}_{b,t}\right) +\frac{f_{2t}p_{t}\left(
1-p_{t}\right) i_{2t}\sigma +f_{3t}\left( 1-p_{t}\right) ^{2}i_{3t}\sigma }{2Q\left(
1-p_{t+1}\right) }
\end{equation}
\end_inset
allowing the calculation,
\begin_inset Formula
\begin{eqnarray}
\Delta d_{t+1} & = & \left( \bar{A}_{B,t+1}-\bar{A}_{b,t+1}\right) -\left(
\bar{A}_{B,t}-\bar{A}_{b,t}\right) \nonumber \\
& = & \frac{f_{2t}p_{t}\left( 1-p_{t}\right) }{2Qp_{t+1}}\left(
\bar{A}_{B,t}-\bar{A}_{b,t}\right) +\frac{f_{1t}p_{t}^{2}i_{1t}\sigma
+f_{2t}p_{t}\left( 1-p_{t}\right) i_{2t}\sigma }{2Qp_{t+1}}\nonumber \\
& & -\frac{f_{2t}p_{t}\left( 1-p_{t}\right) }{2Q\left( 1-p_{t+1}\right)
}d_{t}-\frac{f_{2t}p_{t}\left( 1-p_{t}\right) i_{2t}\sigma +f_{3t}\left(
1-p_{t}\right) ^{2}i_{3t}\sigma }{2Q\left( 1-p_{t+1}\right) }\nonumber \\
& = & \frac{f_{2}p_{t}\left( 1-p_{t}\right) }{2Qp_{t+1}\left( 1-p_{t+1}\right)
}d_{t}\nonumber \\
& & +\frac{p_{t}^{2}\left( 1-p_{t+1}\right) z_{1t}+p_{t}\left( 1-p_{t}\right)
z_{2t}+\left( 1-p_{t}\right) ^{2}p_{t+1}z_{3t}}{2Qp_{t+1}\left( 1-p_{t+1}\right)
}\sigma \label{dp3dt1}
\end{eqnarray}
\end_inset
\layout Standard
Also note that [
\begin_inset LatexCommand \ref{diseqABrule}
\end_inset
] and [
\begin_inset LatexCommand \ref{diseqABrule}
\end_inset
] can be written as
\layout Standard
\begin_inset Formula
\begin{eqnarray}
\bar{A}_{B,T} & = & \bar{A}_{B,0}+\sum _{t=0}^{T-1}\frac{f_{2t}p_{t}\left(
1-p_{t}\right) }{2Qp_{t+1}}\left( \bar{A}_{B,t}-\bar{A}_{b,t}\right)
+\frac{f_{1t}p_{t}^{2}i_{1t}\sigma +f_{2t}p_{t}\left( 1-p_{t}\right) i_{2t}\sigma
}{2Qp_{t+1}}\nonumber \\
& = & \bar{A}_{B,0}+\sum ^{T-1}_{t=0}\Delta \bar{A}_{B,t+1}\label{diseqABassum}
\end{eqnarray}
\end_inset
and
\begin_inset Formula
\begin{equation}
\label{diseqAbassum}
\bar{A}_{b,T}=\bar{A}_{b,0}+\sum _{t=0}^{T-1}\Delta \bar{A}_{b,t+1}
\end{equation}
\end_inset
Which means that as in the prior case, changes
\latex latex
\backslash
jd{
\latex default
in the average polygenic value
\latex latex
}
\latex default
are permanent.
However, as [
\begin_inset LatexCommand \ref{diseqrawobjfn}
\end_inset
] depends upon
\begin_inset Formula \( p_{t} \)
\end_inset
, the value of
\begin_inset Formula \( \Delta \bar{A}_{B,t} \)
\end_inset
is not the same in all future
\latex latex
\backslash
so{
\latex default
equations
\latex latex
}
\backslash
jd{
\latex default
generations
\latex latex
}
\latex default
, making calculations such as [**], which calculates the present value of
a change, impossible.
\begin_float margin
\layout Standard
write prior expresson above
\end_float
\layout Standard
Instead, consider
\begin_inset Formula
\begin{eqnarray}
\bar{A}_{t+1}-\bar{A}_{t} & = & 2p_{t+1}\bar{A}_{Bt+1}+2\left( 1-p_{t+1}\right)
\bar{A}_{b,t+1}-2p_{t}\bar{A}_{B,t}-2\left( 1-p_{t}\right) \bar{A}_{b,t}\nonumber \\
& = & 2\bar{A}_{b,t+1}+2p_{t+1}\left( \bar{A}_{B,t+1}-\bar{A}_{b,t+1}\right)
-2\bar{A}_{b,t}-2p_{t}\left( \bar{A}_{B,t}-\bar{A}_{b,t}\right) \nonumber \\
& = & 2\left( \bar{A}_{b,t+1}-\bar{A}_{b,t}\right)
+2p_{t+1}d_{t+1}-2p_{t}d_{t}\nonumber \\
& = & 2\left( \frac{f_{2t}p_{t}\left( 1-p_{t}\right) }{2Q\left( 1-p_{t+1}\right)
}d_{t}+\frac{f_{2t}p_{t}\left( 1-p_{t}\right) i_{2t}\sigma +f_{3t}\left(
1-p_{t}\right) ^{2}i_{3t}\sigma }{2Q\left( 1-p_{t+1}\right) }\right) \nonumber \\
& & +2p_{t+1}d_{t+1}-2p_{t}d_{t}\nonumber \\
& = & \frac{f_{2t}p_{t}\left( 1-p_{t}\right) \left[ d_{t}+i_{2t}\sigma \right]
+f_{3t}\left( 1-p_{t}\right) ^{2}i_{3t}\sigma }{Q\left( 1-p_{t+1}\right)
}+2p_{t+1}d_{t+1}-2p_{t}d_{t}\nonumber \\
& = & \frac{p_{t}\left( 1-p_{t}\right) \left[ f_{2t}d_{t}+z_{2t}\sigma \right]
+\left( 1-p_{t}\right) ^{2}z_{3t}\sigma }{Q\left( 1-p_{t+1}\right)
}+2p_{t+1}d_{t+1}-2p_{t}d_{t}\label{diseqArule}
\end{eqnarray}
\end_inset
which
\latex latex
\backslash
so{
\latex default
, while not pretty,
\latex latex
}
\latex default
is written entirely in terms of the state variables
\begin_inset Formula \( p_{t} \)
\end_inset
and
\begin_inset Formula \( d_{t} \)
\end_inset
, and fractions which are functions of these choice variables.
Thus [
\begin_inset LatexCommand \ref{diseqArule}
\end_inset
] can be used to write the present value of
\latex latex
\backslash
so{
\latex default
a change
\latex latex
}
\latex default
\latex latex
\backslash
jd{
\latex default
the change in state
\latex latex
}
\latex default
as
\begin_inset Formula
\begin{equation}
pv=\frac{2\left( 1-r\right) }{r}\left[ \bar{A}_{t+1}-\bar{A}_{t}\right]
\end{equation}
\end_inset
\layout Subsection
The State
\layout Standard
The state at any time will be described by the pair
\begin_inset Formula \( \left( p_{t},d_{t}\right) \)
\end_inset
, where
\begin_inset Formula \( d_{t} \)
\end_inset
is the disequilibrium at
\latex latex
\backslash
so{
\latex default
the
\latex latex
}
\latex default
time
\latex latex
\backslash
jd{
\latex default
\begin_inset Formula \( t \)
\end_inset
\latex latex
}
\latex default
,
\begin_inset Formula
\begin{equation}
d_{t}=\bar{A}_{B,t}-\bar{A}_{b,t}
\end{equation}
\end_inset
Rather than calculating
\begin_inset Formula \( f_{1t} \)
\end_inset
to maximize the polygenic gain, the successor states
\latex latex
\backslash
jd{
\latex default
\begin_inset Formula \( p_{t+1} \)
\end_inset
and
\begin_inset Formula \( d_{t+1} \)
\end_inset
\latex latex
}
\latex default
are chosen, and the choice made fully dictates
\begin_inset Formula \( f_{t} \)
\end_inset
.
Equations [
\begin_inset LatexCommand \ref{dp2prule}
\end_inset
], [
\begin_inset LatexCommand \ref{dp3dt1}
\end_inset
], and [
\begin_inset LatexCommand \ref{dp2qcons}
\end_inset
] constitute a system of three equations in the three unknowns
\begin_inset Formula \( f_{t} \)
\end_inset
, which can be solved by a library routine.
As a practical matter, [
\begin_inset LatexCommand \ref{dp2qcons}
\end_inset
] is used to eliminate
\begin_inset Formula \( f_{3t} \)
\end_inset
from the system to reduce computation.
\layout Subsection
Mass selection With Gametic Phase Disequilibrium
\layout Standard
\latex latex
\backslash
so{
\latex default
While
\latex latex
}
\backslash
jd{
\latex default
When
\latex latex
}
\latex default
igoring disequilibrium, all genotypes had the same polygenic distribution,
and mass selection truncation points
\begin_inset Formula \( x_{t} \)
\end_inset
were calculated by setting
\begin_inset Formula
\begin{equation}
x_{1t}-\frac{a}{\sigma }=x_{2t}=x_{3t}+\frac{a}{\sigma }
\end{equation}
\end_inset
This is no longer the case under disequilibrium; the new rule is
\begin_inset Formula
\begin{equation}
x_{1t}+\left( 2\bar{A}_{B,t}-\frac{a}{\sigma }\right) =x_{2t}+\left(
\bar{A}_{B,t}+\bar{A}_{b,t}\right) =x_{3t}+2\bar{A}_{b,t}+\frac{a}{\sigma }
\end{equation}
\end_inset
which can be rewritten as
\begin_inset Formula
\begin{equation}
x_{1t}+\left( 2\bar{A}_{b,t}+2d_{t}-\frac{a}{\sigma }\right) =x_{2t}+\left(
2\bar{A}_{b,t}+d_{t}\right) =x_{3t}+\left( 2\bar{A}_{b,t}+\frac{a}{\sigma }\right)
\end{equation}
\end_inset
or
\begin_inset Formula
\begin{equation}
x_{1t}+\left( d_{t}-\frac{a}{\sigma }\right) =x_{2t}=x_{3t}-\left(
d_{t}-\frac{a}{\sigma }\right)
\end{equation}
\end_inset
\layout Subsection
Translation from state to choice variables
\layout Standard
Given that the problem is to be solved with state variables rather than
choice variables, it is necessary to have a mechanism allowing translation
from state transitions to the corresponding choice variables, both to report
the results and to check for violations of possible states.
\layout Standard
Given that
\begin_inset Formula \( f_{1t} \)
\end_inset
has already been eliminated from [
\begin_inset LatexCommand \ref{diseqArule}
\end_inset
], it is useful to solve [
\begin_inset LatexCommand \ref{dp2prule}
\end_inset
] for
\begin_inset Formula \( f_{1t} \)
\end_inset
,
\begin_inset Formula
\begin{equation}
f_{1t}=\frac{Qp_{t+1}-f_{2t}p_{t}\left( 1-p_{t}\right) }{p_{t}^{2}}
\end{equation}
\end_inset
which can be used to solve [
\begin_inset LatexCommand \ref{dp2qcons}
\end_inset
] for
\begin_inset Formula \( f_{3t} \)
\end_inset
,
\begin_inset Formula
\begin{eqnarray}
f_{3t} & = & \frac{Q-\frac{Qp_{t+1}-f_{2t}p_{t}\left( 1-p_{t}\right)
}{p_{t}^{2}}p_{t}^{2}-2p_{t}\left( 1-p_{t}\right) f_{2t}}{\left( 1-p_{t}\right)
^{2}}\nonumber \\
& = & \frac{Q-Qp_{t+1}+f_{2t}p_{t}\left( 1-p_{t}\right) -2p_{t}\left( 1-p_{t}\right)
f_{2t}}{\left( 1-p_{t}\right) ^{2}}\nonumber \\
& = & \frac{Q\left( 1-p_{t+1}\right) -p_{t}\left( 1-p_{t}\right) f_{2t}}{\left(
1-p_{t}\right) ^{2}}
\end{eqnarray}
\end_inset
which is in turn csubstituted into [
\begin_inset LatexCommand \ref{diseqArule}
\end_inset
]
\begin_inset Formula
\begin{eqnarray}
0 & = & f_{2}p_{t}\left( 1-p_{t}\right) \frac{d_{t}}{\sigma }\nonumber \\
& & +p_{t}^{2}\left( 1-p_{t+1}\right) z\left( \frac{Qp_{t+1}-f_{2t}p_{t}\left(
1-p_{t}\right) }{p_{t}^{2}}\right) \nonumber \\
& & +p_{t}\left( 1-p_{t}\right) z\left( f_{2t}\right) \nonumber \\
& & +\left( 1-p_{t}\right) ^{2}p_{t+1}z\left( \frac{Q\left( 1-p_{t+1}\right)
-p_{t}\left( 1-p_{t}\right) f_{2t}}{\left( 1-p_{t}\right) ^{2}}\right) \nonumber \\
& & -\frac{\left( d_{t+1}-d_{t}\right) }{\sigma }\left( 2Qp_{t+1}\left(
1-p_{t+1}\right) \right)
\end{eqnarray}
\end_inset
which can be solved by a fast numeric routine
\layout Subsection
Translation from state to choice variables (new)
\layout Standard
While working with the state variables
\begin_inset Formula \( d \)
\end_inset
and
\begin_inset Formula \( p \)
\end_inset
is sufficient to describe the
\emph on
behavior
\emph default
of the system, it is not enough to quantitatively
\emph on
evaluate
\emph default
a path: the original
\begin_inset Formula \( f_{t} \)
\end_inset
,
\begin_inset Formula \( x_{t} \)
\end_inset
and
\begin_inset Formula \( z_{t} \)
\end_inset
are needed in equations such as [
\begin_inset LatexCommand \ref{diseqArule}
\end_inset
] to determine the value of the proposed state to the breeder.
Equations [
\begin_inset LatexCommand \ref{diseqArule}
\end_inset
], [
\begin_inset LatexCommand \ref{diseqdrule}
\end_inset
], and [
\begin_inset LatexCommand \ref{dp2qcons}
\end_inset
] are three equations in the three choice variables, determining them exactly.
While it is not possible to write an explicit, closed form solution, it
is possible to reduce the equaitons to a single variable in a single unknown,
and then solve with a fast numeric routine.
\layout Standard
As
\begin_inset Formula \( f_{1t} \)
\end_inset
has already been eliminated from [
\begin_inset LatexCommand \ref{diseqArule}
\end_inset
], and
\begin_inset Formula \( f_{3t} \)
\end_inset
has generally been treated as an artifact of the choice of
\begin_inset Formula \( f_{1t} \)
\end_inset
and
\begin_inset Formula \( f_{2t} \)
\end_inset
,
\begin_inset Formula \( f_{2t} \)
\end_inset
seems to be the natural choice.
Manipulating [
\begin_inset LatexCommand \ref{dp2prule}
\end_inset
] yields
\begin_inset Formula
\begin{equation}
f_{1t}=\frac{Qp_{t+1}-f_{2t}p_{t}\left( 1-p_{t}\right) }{p_{t}^{2}}
\end{equation}
\end_inset
which can in turn be used in [
\begin_inset LatexCommand \ref{dp2qcons}
\end_inset
] to get
\begin_inset Formula
\begin{eqnarray}
f_{3t} & = & \frac{Q-\frac{Qp_{t+1}-f_{2t}p_{t}\left( 1-p_{t}\right)
}{p_{t}^{2}}p_{t}^{2}-2p_{t}\left( 1-p_{t}\right) f_{2t}}{\left( 1-p_{t}\right)
^{2}}\nonumber \\
& = & \frac{Q-Qp_{t+1}+f_{2t}p_{t}\left( 1-p_{t}\right) -2p_{t}\left( 1-p_{t}\right)
f_{2t}}{\left( 1-p_{t}\right) ^{2}}\nonumber \\
& = & \frac{Q\left( 1-p_{t+1}\right) -p_{t}\left( 1-p_{t}\right) f_{2t}}{\left(
1-p_{t}\right) ^{2}}
\end{eqnarray}
\end_inset
both of which are placed into [
\begin_inset LatexCommand \ref{diseqdrule}
\end_inset
], [
\begin_inset LatexCommand \ref{diseqdrule}
\end_inset
],
\begin_inset Formula
\begin{eqnarray}
0 & = & d_{t}-d_{t+1}-\frac{1}{2Q}f_{2t}\frac{p_{t}\left( 1-p_{t}\right)
}{p_{t+1}\left( 1-p_{t+1}\right) }d_{t}\nonumber \\
& & +\frac{1}{2Q}\left[ \frac{p_{t}^{2}}{p_{t+1}}z\left(
\frac{Qp_{t+1}-f_{2t}p_{t}\left( 1-p_{t}\right) }{p_{t}^{2}}\right) +p_{t}\left(
1-p_{t}\right) \left[ \frac{1}{p_{t+1}}-\frac{1}{1-p_{t+1}}\right] z\left(
f_{2t}\right) \right. \nonumber \\
& & \left. -\frac{\left( 1-p_{t}\right) ^{2}}{\left( 1-p_{t+1}\right) }z\left(
\frac{Q\left( 1-p_{t+1}\right) -p_{t}\left( 1-p_{t}\right) f_{2t}}{\left(
1-p_{t}\right) ^{2}}\right) \right] \sigma
\end{eqnarray}
\end_inset
which has a derivative of
\begin_inset Formula
\begin{eqnarray*}
der_{f_{2}} & = & -\frac{1}{2Q}\frac{p_{t}\left( 1-p_{t}\right) }{p_{t+1}\left(
1-p_{t+1}\right) }d_{t}\\
& & +\frac{1}{2Q}\left[ \frac{p_{t}^{2}}{p_{t+1}}x\left(
\frac{Qp_{t+1}-f_{2t}p_{t}\left( 1-p_{t}\right) }{p_{t}^{2}}\right) \frac{-\left(
1-p_{t}\right) }{p_{t}}\right. \\
& & +p_{t}\left( 1-p_{t}\right) \left[ \frac{1}{p_{t+1}}-\frac{1}{1-p_{t+1}}\right]
x\left( f_{2t}\right) \\
& & \left. -\frac{\left( 1-p_{t}\right) ^{2}}{\left( 1-p_{t+1}\right) }x\left(
\frac{Q\left( 1-p_{t+1}\right) -p_{t}\left( 1-p_{t}\right) f_{2t}}{\left(
1-p_{t}\right) ^{2}}\right) \frac{-p_{t}}{1-p_{t}}\right] \sigma \\
& = & -\frac{\sigma }{2Q}\frac{p_{t}\left( 1-p_{t}\right) }{p_{t+1}\left(
1-p_{t+1}\right) }\left[ \frac{d_{t}}{\sigma }+\left( 1-p_{t+1}\right) x_{1}-\left(
1-2p_{t+1}\right) x\left( f_{2t}\right) -p_{t+1}x_{3}\right]
\end{eqnarray*}
\end_inset
\layout Standard
\begin_inset Formula
\begin{eqnarray}
& = & \frac{1}{2Q}\frac{p_{t}\left( 1-p_{t}\right) }{p_{t+1}\left( 1-p_{t+1}\right)
}d_{t}\nonumber \\
& & -\frac{1}{2Q}\left[ \frac{p_{t}^{2}}{p_{t+1}}x\left(
\frac{Qp_{t+1}-f_{2t}p_{t}\left( 1-p_{t}\right) }{p_{t}^{2}}\right)
\frac{1-p_{t}}{p_{t}}\right. \nonumber \\
& & +p_{t}\left( 1-p_{t}\right) \left[ \frac{1}{p_{t+1}}-\frac{1}{1-p_{t+1}}\right]
x\left( f_{2t}\right) \nonumber \\
& & \left. -\frac{\left( 1-p_{t}\right) ^{2}}{\left( 1-p_{t+1}\right) }x\left(
\frac{Q\left( 1-p_{t+1}\right) -p_{t}\left( 1-p_{t}\right) f_{2t}}{\left(
1-p_{t}\right) ^{2}}\right) \frac{p_{t}}{1-p_{t}}\right] \sigma \nonumber \\
& = & \frac{1}{2Q}\frac{p_{t}\left( 1-p_{t}\right) }{p_{t+1}\left( 1-p_{t+1}\right)
}d_{t}\nonumber \\
& & -\frac{1}{2Q}\left[ \frac{p_{t}\left( 1-p_{t}\right) }{p_{t+1}}x\left(
\frac{Qp_{t+1}-f_{2t}p_{t}\left( 1-p_{t}\right) }{p_{t}^{2}}\right) \right. \nonumber
\\
& & +p_{t}\left( 1-p_{t}\right) \left[ \frac{1}{p_{t+1}}-\frac{1}{1-p_{t+1}}\right]
x\left( f_{2t}\right) \nonumber \\
& & \left. -\frac{p_{t}\left( 1-p_{t}\right) }{\left( 1-p_{t+1}\right) }x\left(
\frac{Q\left( 1-p_{t+1}\right) -p_{t}\left( 1-p_{t}\right) f_{2t}}{\left(
1-p_{t}\right) ^{2}}\right) \right] \sigma \nonumber \\
& = & \frac{p_{t}\left( 1-p_{t}\right) }{2Q}\sigma \left[ \frac{\frac{d_{t}}{\sigma
}}{p_{t+1}\left( 1-p_{t+1}\right) }-\frac{x\left( \frac{Qp_{t+1}-f_{2t}p_{t}\left(
1-p_{t}\right) }{p_{t}^{2}}\right) }{p_{t+1}}\right. \nonumber \\
& & \left. -\left[ \frac{1}{p_{t+1}}-\frac{1}{1-p_{t+1}}\right] x\left(
f_{2t}\right) +\frac{x\left( \frac{Q\left( 1-p_{t+1}\right) -p_{t}\left(
1-p_{t}\right) f_{2t}}{\left( 1-p_{t}\right) ^{2}}\right) }{1-p_{t+1}}\right]
\nonumber \\
& = & \frac{p_{t}\left( 1-p_{t}\right) }{p_{t+1}\left( 1-p_{t+1}\right) }\frac{\sigma
}{2Q}\left[ \frac{d_{t}}{\sigma }-\left( 1-p_{t+1}\right) x\left(
\frac{Qp_{t+1}-f_{2t}p_{t}\left( 1-p_{t}\right) }{p_{t}^{2}}\right) \right. \nonumber
\\
& & \left. -x\left( f_{2t}\right) +p_{t+1}x\left( \frac{Q\left( 1-p_{t+1}\right)
-p_{t}\left( 1-p_{t}\right) f_{2t}}{\left( 1-p_{t}\right) ^{2}}\right) \right]
\end{eqnarray}
\end_inset
\layout Subsection
Is
\begin_inset Formula \( \Delta \bar{A}_{t+1} \)
\end_inset
monotone in
\begin_inset Formula \( d_{t+1} \)
\end_inset
?
\layout Standard
Differentiating [
\begin_inset LatexCommand \ref{diseqArule}
\end_inset
],
\begin_inset Formula
\begin{eqnarray*}
\frac{\partial \Delta \bar{A}_{t+1}}{\partial d_{t+1}} & = & \frac{p_{t}\left(
1-p_{t}\right) d_{t}}{Q\left( 1-p_{t+1}\right) }\frac{\partial f_{2t}}{\partial
d_{t+1}}+\frac{\left( 1-p_{t}\right) ^{2}\sigma }{Q\left( 1-p_{t+1}\right)
}\frac{\partial z_{3t}}{\partial d_{t+1}}+2p_{t+1}\\
& = & \frac{p_{t}\left( 1-p_{t}\right) d_{t}}{Q\left( 1-p_{t+1}\right)
}\frac{1}{\frac{\partial d_{t+1}}{\partial f_{2t}}}+\frac{\left( 1-p_{t}\right)
^{2}\sigma }{Q\left( 1-p_{t+1}\right) }\frac{\partial z_{3t}}{\partial
f_{3t}}\frac{\partial f_{3t}}{\partial f_{2t}}\frac{1}{\frac{\partial
d_{t+1}}{\partial f_{2t}}}+2p_{t+1}\\
& = & \frac{p_{t}\left( 1-p_{t}\right) d_{t}}{Q\left( 1-p_{t+1}\right)
}\frac{1}{\frac{\partial d_{t+1}}{\partial f_{2t}}}+\frac{\left( 1-p_{t}\right)
^{2}\sigma }{Q\left( 1-p_{t+1}\right) }\frac{\partial z_{3t}}{\partial
f_{3t}}\frac{\partial f_{3t}}{\partial f_{2t}}\frac{1}{\frac{\partial
d_{t+1}}{\partial f_{2t}}}+2p_{t+1}
\end{eqnarray*}
\end_inset
\layout Standard
\latex latex
\backslash
pagebreak
\layout Subsection
N-dimensional search rules
\layout Standard
An algorithm that can solve
\begin_inset Formula \( n \)
\end_inset
dimensions rather than a fixed number is more valuable and easily maintained
than a set of algorithms for assorted dimensions.
Furthermore, it is easier to write than an algorithm for any fixed
\begin_inset Formula \( n \)
\end_inset
larger than two.
Accordingly, the focus will be on the algorithm of arbitrary dimension.
\layout Standard
To solve [
\begin_inset LatexCommand \pageref{dp2finalobj}
\end_inset
], steps were made in one direction or the other until a gain was no longer
found.
This process may be used iteratively to solve multiple dimensions.
\family typewriter
bestval()
\family default
is initially called with no location information, causing it to start at
the first variable.
It evaluates this variable as before, but rather than directly calling
the next generation, it calls the same generation, but with the current
iteration value.
\layout Standard
When
\family typewriter
bestVal()
\family default
receives an array of positions of rank
\begin_inset Formula \( n \)
\end_inset
, it queries the next generation.
\layout Section
Four Dimensions: Differential Selection by Gender
\layout Section
Unknown Infinite Horizon
\layout Section
Variable variance
\layout Standard
The actual variance is binomial, for large enough
\begin_inset Formula \( n \)
\end_inset
that it is adequately approximated by the normal.
\layout Standard
Redefine model.
\begin_inset Formula \( n \)
\end_inset
is the number of minor gene sites, and
\begin_inset Formula \( q \)
\end_inset
is the minor gene frequency.
Letting
\begin_inset Formula \( \pi \)
\end_inset
and
\begin_inset Formula \( \theta \)
\end_inset
be the draw for each,
\begin_inset Formula
\begin{eqnarray*}
\pi & \in & \left\{ 0,1,2\right\} \\
\theta & \in & \left\{ 0,1,\ldots 2n\right\}
\end{eqnarray*}
\end_inset
and the value of an animal is
\begin_inset Formula
\[
\pi a+\theta b-\left( a+nq_{0}b\right) \]
\end_inset
\layout Chapter
\latex latex
\backslash
sout{moved to appendix}
\layout Chapter
Results
\begin_inset LatexCommand \label{Results}
\end_inset
\layout Section
Simple Finite Generations
\layout Subsubsection
Simple Newton Raphson
\begin_inset LatexCommand \label{NR/SFG}
\end_inset
\layout Section
Discounted Finite Generations
\layout Section
Infinite Horizon
\layout Chapter
Conclusions
\layout Standard
\latex latex
\backslash
appendix
\layout Chapter
Glossary
\layout Section
Terms
\layout Subsubsection*
Breeding Value
\layout Standard
The average effect of all genes that a parent passes on to offspring..
\layout Subsubsection*
Diploid
\layout Standard
Having two chromosomes.
All animals, and many plants, are diploid.
\layout Subsubsection*
Genetic Phase Disequilibrium
\layout Standard
When different truncation points are used for the genotypes, the result
is a different truncation point for the polygenes in each group.
In the next generation, the polygenes will have different means and variances
in the different group.
This creates a negative correlation between the major and polygenes known
as Gametic Phase Disequilibrium.
\layout Subsubsection*
Genotype
\layout Standard
Classification by the presence of the gene.
E.g., aa, aA, and AA.
\layout Subsubsection*
Genotype
\layout Standard
The actual genetic status of the organism for a given locus.
For example,
\begin_inset Formula \( Bb \)
\end_inset
.
\layout Subsubsection*
Heterozygote
\layout Standard
In a diploid organism, a heterozygote for a given locus has a two different
alleles at the locus in question.
\layout Subsubsection*
Heritability
\layout Standard
The fraction of phenotypic variation in a trait that is due to genetics.
\layout Subsubsection*
Homozygote
\layout Standard
An organism with two of the same allele at the locus in question.
\layout Subsubsection*
Locus
\layout Standard
A point on a chromosome where a gene is located.
\layout Subsubsection*
Major Gene
\layout Standard
A gene with a large effect.
It is assumed that the major gene can be identified, by QTL or other methods.
\layout Subsubsection*
Mass Selection
\layout Standard
Also phenotypic selection.
Organisms are selected to reproduce based solely upon their own phenotypic
value for the trait.
\layout Subsubsection*
Phenotype
\layout Standard
The observed trait.
For example, the weight of an animal.
\layout Subsubsection*
Polygenes
\layout Standard
Polygenes cannot be identified, but are seen only by their combined effect
on phenotype.
It is assumed that each of the polygenes has a small effect compared to
the whole and to the major gene.
\layout Subsubsection*
Qualitative trait
\layout Standard
A trait that takes on only one of a finite set of possibilities.
\layout Subsubsection*
Quantitative Trait
\layout Standard
A quantitative trait is one which takes a quantitative rather than qualitative
value.
Mendel's peas were qualitatively either wrinkled or not; they were one
of the two types.
A quantitative trait would instead be a measure of the height of the peas:
a value in the range of three to six inches, for example, with the intermediate
values being possible.
\layout Subsubsection*
QTL
\layout Standard
Quantitative trait locus, the locus that controls or affects a quantitative
trait.
\layout Subsubsection*
Genotypic Selection
\layout Standard
Genotypic selection considers both the phenotypic and genotypic values.
a value of
\begin_inset Formula \( I=g+h^{2}(P-g) \)
\end_inset
is used, where
\begin_inset Formula \( P \)
\end_inset
is the phenotypic value,
\begin_inset Formula \( g \)
\end_inset
the genotypic value, and
\begin_inset Formula \( h^{2} \)
\end_inset
the heritability.
\layout Subsubsection*
Truncation Point.
\layout Standard
A cutoff point on the selection criterion for selection.
All creatures above this point breed, and those below do not.
\layout Section
Variable Names and Definitions
\layout Standard
The following variables have the following meaning unless otherwise specified:d
\layout Standard
\added_space_top 0.3cm \added_space_bottom 0.3cm \align center \LyXTable
multicol5
12 2 0 0 -1 -1 -1 -1
1 1 0 0
1 0 0 0
1 0 0 0
1 0 0 0
1 0 0 0
1 0 0 0
1 0 0 0
1 0 0 0
1 0 0 0
1 0 0 0
1 0 0 0
1 1 0 0
8 1 0 "" ""
8 1 1 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
0 8 1 0 0 0 0 "" ""
Variable
\newline
Meaning
\newline
\begin_inset Formula \( g \)
\end_inset
\newline
genotypic value for the identified major gene
\newline
\begin_inset Formula \( h^{2} \)
\end_inset
\newline
heritability of the trait
\newline
\begin_inset Formula \( P \)
\end_inset
\newline
the phenotypic value
\newline
\begin_inset Formula \( m \)
\end_inset
\newline
an indicator for genotype
\newline
\begin_inset Formula \( A \)
\end_inset
\newline
polygenic breeding value
\newline
\begin_inset Formula \( \hat{A} \)
\end_inset
\newline
estimated polygenic breeding value
\newline
\begin_inset Formula \( b_{mt} \)
\end_inset
\newline
weight used for genotype
\begin_inset Formula \( m \)
\end_inset
selected in generation
\begin_inset Formula \( t \)
\end_inset
\newline
\begin_inset Formula \( a \)
\end_inset
\newline
the value of each allele of the major gene
\newline
\begin_inset Formula \( p_{t} \)
\end_inset
\newline
the frequency of the major gene in generation
\begin_inset Formula \( t \)
\end_inset
\newline
\begin_inset Formula \( q_{mt} \)
\end_inset
\newline
the fraction of the population in generation
\begin_inset Formula \( t \)
\end_inset
of type
\begin_inset Formula \( m \)
\end_inset
\newline
\begin_inset Formula \( x_{mt} \)
\end_inset
\newline
the truncation point for genotype
\begin_inset Formula \( m \)
\end_inset
in generation
\begin_inset Formula \( t \)
\end_inset
\layout Section
The Assumptions
\layout Standard
This work is done primarily in the limiting case of a very large population,
with a vary large number of genes.
It is assumed that the breeding value can be written as an additive function
of the major genes and of the sum of the interactions of the polygenes,
and the statistical distribution of the polygenes is known.
For the baseline case, there is a single major gene, and each copy of the
gene adds directly to a normally distributed value for the polygenes.
A heterozygote receives a value of
\begin_inset Formula \( 0 \)
\end_inset
, while the homozygotes receive
\begin_inset Formula \( \pm a \)
\end_inset
.
The variance of the polygenic breeding value is known, the distribution
is normal, and it is initially normalized at 0, and equal fractions of
males and females are chosen.
\begin_float footnote
\layout Standard
If this were not done, the solution would be to take few enough males to
impregnate all the females, and the effects of inbreeding would have to
be considered.
\end_float
\layout Standard
The decrease in variance of the polygenic breeding value due to selection
and phase disequilibrium are disregarded in the simplest case.
Also, due to random mating of those chosen to reproduce, the major gene
will always be in equilibrium: all genotypes have the same
\begin_inset Formula \( p_{t} \)
\end_inset
, and so an organism has the same chance of getting the major gene from
a parent regardless of the genotype to which the parent belonged.
\layout Chapter
dp2 program
\layout Bibliography
\bibitem [Dekkers 98]{Dekkers}
Dekkers paper.
\layout Bibliography
\bibitem [Varian]{varian}
Varian
\layout Bibliography
\bibitem [Vertical Coordination]{Vertical Coordination}
Vertical Coordination and Consumer Welfare: The Case of the Pork Industry,
USDA AER #753, August, 1997
\layout Bibliography
\bibitem [Bulmer]{Bulmer}
Bulmer, The Mathematical Theory of Quantitative Genetics, Clarendon Press,
1980
\layout Bibliography
\bibitem [Rothschild]{Rothschild}
Rothschild,***
\layout Bibliography
\bibitem [Gibsen]{Gibsen}
Gibsen, **, 1994
\layout Bibliography
\bibitem [IQG]{iqg}
Falconer & Mackay,Introduction to Quantitative Genetics, Fourth Edition,Longman,
1966.
\layout Standard
\bibitem {4}
\begin_inset LatexCommand \printindex{}
\end_inset
\the_end
