The attached .R file should demonstrate what Ross was talking about.

The cardinality constraint is directly supported by the stochastic
global solvers 'DEoptim' and 'random', as described in the
documentation.  

It can also be formulated as a mixed integer linear problem for certain
objective functions, but because you're looknig for the Markowitz
mean/variance portfolio, you have a quadratic problem, and can't
formulate it using an MILP.

The attached file fixes the minimum position box constraint to 0, which
aalows all solvers to converge.  

pso still doesn't get a good solution, but this solution may be improved
by increasing the maximum number of iterations via the parameter
'maxit'. After increasing maxit, it gets closer to the solutions
returned by 'random' and 'DEoptim'.

Both 'random' and 'DEoptim' produce similar solutions which meet the
constraints and objectives, but they are not identical, and change over
multiple runs.  (This is stochastic optimization, after all, so the only
way to get the same answer would be to artificially, and incorrectly,
set the random seed.).

This suggests two things:

1> that there are not enough random portfolios (for 'random') or that
the population size and number of generations are insufficient to find
the global solution (for DEoptim).

2> that the problem may still be somewhat over constrained, most likely
by the max weight box constraint.  You might want to use random
portfolios and portfolio sets to compare and plot the difference between
the unconstrained search space and the constrained search space to sort
out whether this is really what you want to do.

Regards,

Brian

-- 
Brian G. Peterson
http://braverock.com/brian/
Ph: 773-459-4973
IM: bgpbraverock


On Tue, 2016-09-20 at 07:18 -0500, Ross Bennett wrote:
> Hi Abhay,
> 
> The cardinality constraint is not directly support by the pso solver so we
> have to implement that constraint as a penalized objective. Also note that
> your box constraint is greater than 0, so the box constraint and position
> limit constraint are fighting each other (i.e your problem is
> overconstrained).
> 
> I recommend using DEoptim or random portfolios as a solver when dealing
> with position limit constraints. Position limit constraints are supported
> directly by the algorithm in the random portfolios 'sample' method. We use
> the mapping function supported by DEoptim to handle more complex
> constraints such as position limit when using DEoptim as the solver.
> 
> Hope that helps.
> 
> Ross
> 
> 
> On Mon, Sep 19, 2016 at 10:25 AM, Abhay Bhadani <abhad...@gmail.com> wrote:
> 
> > Thanks, Brian!
> >
> > I implemented the following:
> > ------------------------------------------------------------
> > ----------------------------------
> > data("edhec")
> > returns <- edhec[,1:12]
> > colnames(returns) <-
> > c("CA","CTAG","DS","EM","EN","ED","FIA","GMLS","MA","RV","SS","FF")
> > print(head(returns,5))
> > fund.names <- colnames(returns)
> >
> >
> > #Giving Portfolio Specifications
> > pspec <- portfolio.spec(assets=fund.names)
> > print.default(pspec)
> >
> > #Adding Constraints
> > #Full investment constraint: sum of all x_i is 1
> > pspec <- add.constraint(portfolio= pspec, type =
> > "weight_sum",min=0.99,max=1.01)
> >
> > #Box constraint: value of x_i varies between 0.2 to 0.8
> > pspec <- add.constraint(portfolio= pspec, type="box", min = 0.01, max =
> > 0.25)
> >
> > #Cardinality constraint
> > pspec <- add.constraint(portfolio= pspec, type="position_limit",max_pos =
> > 6,enabled=TRUE)
> >
> > #Adding Objective
> > pspec <- add.objective(portfolio=pspec, type="risk", name="var")
> > pspec <- add.objective(portfolio=pspec, type="return", name="mean")
> >
> > opt_meanvar <- optimize.portfolio(R=returns,portfolio = pspec,
> >                                   optimize_method="pso", trace=TRUE)
> >
> > ------------------------------------------------------------
> > ------------------------------
> >
> > Results:
> >
> > Optimal Weights:
> >     CA   CTAG     DS     EM     EN     ED    FIA   GMLS     MA     RV
> > SS     FF
> > 0.1319 0.0551 0.0312 0.1312 0.1127 0.1467 0.0641 0.0218 0.0805 0.1101
> > 0.0296 0.0952
> >
> > _____________________________________________________
> >
> > I obtained weights for all 12 assets. That's why I was not sure whether
> > position_limit constraint is same as cardinality constraint.
> >
> >
> > On Mon, Sep 19, 2016 at 8:32 PM, Brian G. Peterson <br...@braverock.com>
> > wrote:
> >
> > > On Mon, 2016-09-19 at 20:22 +0530, Abhay Bhadani wrote:
> > > > I just started exploring PortfolioAnalytics package.
> > > >
> > > > Similar to setting up custom objective functions, is there a way to set
> > > up
> > > > custom constraints too?
> > > >
> > > > I would like to know how to set up cardinality constraint (i.e.,
> > limiting
> > > > number of assets in a portfolio).
> > >
> > > cardinality constraints are already supported via the 'position_limit'
> > > constraint which is an integer constraint limiting the maximum number of
> > > non-zero weight positions in the portfolio.  It may be added like this:
> > >
> > >
> > > pspec <- add.constraint(portfolio=pspec,
> > >                         type="position_limit",
> > >                         max_pos=3,
> > >                         enabled=TRUE)
> > >
> > > assuming that your portfolio specification object is 'pspec'.
> > >
> > > As with other constraint types, this may not be efficiently supported by
> > > all optimization engines. (This is a limitation of the underlying
> > > optimizers/solvers, not of PortfolioAnalytics).
> > >
> > > On a more general note, any constraint may be expressed as an objective
> > > by creating a penalty for violating the constraint.  As noted above,
> > > this may lead to very inefficient or non-converging optimization.
> > >
> > > Regards,
> > >
> > > Brian
> > >
> > > --
> > > Brian G. Peterson
> > > http://braverock.com/brian/
> > > Ph: 773-459-4973
> > > IM: bgpbraverock
> > >
> > >
> > >
> > >
> > >
> >
> >         [[alternative HTML version deleted]]
> >
> > _______________________________________________
> > R-SIG-Finance@r-project.org mailing list
> > https://stat.ethz.ch/mailman/listinfo/r-sig-finance
> > -- Subscriber-posting only. If you want to post, subscribe first.
> > -- Also note that this is not the r-help list where general R questions
> > should go.
> >
> 
>       [[alternative HTML version deleted]]
> 
> _______________________________________________
> R-SIG-Finance@r-project.org mailing list
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> should go.
data("edhec")
returns <- edhec[,1:12]
colnames(returns) <-
  c("CA","CTAG","DS","EM","EN","ED","FIA","GMLS","MA","RV","SS","FF")
print(head(returns,5))
fund.names <- colnames(returns)


#Giving Portfolio Specifications
pspec <- portfolio.spec(assets=fund.names)
print.default(pspec)

#Adding Constraints
#Full investment constraint: sum of all x_i is 1
pspec <- add.constraint(portfolio= pspec, type =
                          "weight_sum",min=0.99,max=1.01)

#Box constraint: value of x_i varies between 0.0 to 0.25
pspec <- add.constraint(portfolio= pspec, type="box", min=0.00, max=0.25)

#Cardinality constraint
pspec <- add.constraint(portfolio= pspec, type="position_limit" , max_pos=6)

#Adding Objective
pspec <- add.objective(portfolio=pspec, type="risk", name="var")
pspec <- add.objective(portfolio=pspec, type="return", name="mean")

# doesn't get very close
opt_meanvar.pso.1  <- optimize.portfolio(R=returns,portfolio = pspec,
                                         optimize_method="pso", trace=TRUE)

# gets closer by increasing thenumber of iterations
opt_meanvar.pso.2  <- optimize.portfolio(R=returns,portfolio = pspec,
                                         optimize_method="pso", trace=TRUE, 
                                         maxit=5000)

# works
opt_meanvar.random <- optimize.portfolio(R=returns,portfolio = pspec, 
                                         optimize_method="random", trace=TRUE)

#works
opt_meanvar.deoptim <- optimize.portfolio(R=returns,portfolio = pspec,
                                         optimize_method="DEoptim", trace=TRUE)

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