Michael, Does this not imply that we could just say sum = 2a + z where a is an integer >=0 and z is binary?
-- Nigel Galloway [email protected] On Wed, Oct 9, 2013, at 06:00 AM, Meketon, Marc wrote: > Michael. This is also very clever. > > Another explanation of your code is the following: > > Variable a is a non-negative integer (and always <= N2hi), b is > binary, z is binary. There are 4 constraints: > (1) sum=2a+b > (2) z>=b-a > (3) z<=b > (4) z<=(1-N2hi/N2lo)b - a/N2lo + N2hi/N2lo > > When sum=1 we must have a=0, b=1. Constraints (2) becomes z >= 1, (3) > becomes z <= 1, and (4) becomes z <=1. Hence z = 1. > > For any sum that is even, a = sum/2 and b=0. Constraint (2) is then > non-binding since b-a <=0 and we know z >=0. Constraint (3) is z <=0. > Constraint (4) (since b=0) is z <= (N2hi - a)/N2lo. Since 0 <= a <= > N2hi, this is a non-binding since constraint (3) is tighter. Hence z=0. > > For any sum that is odd, with sum >= 3, we know that 1 <= a <= N2lo and b > = 1. Constraint (2) is non-binding because b-a <= 0. Constraint (3) > with z <= 1 is non-binding. Constraint (4) becomes z <= 1-a/N2lo. Since > 1 <= a <= N2lo, we know 0 <= 1-a/N2lo < 1, implying z < 1 (strict > inequality), and then the binary constraint forces z=0. > > > -----Original Message----- > From: Michael Hennebry [mailto:[email protected]] > Sent: Tuesday, October 08, 2013 1:43 PM > To: Meketon, Marc > Cc: Nigel Galloway; Andrew Makhorin; [email protected]; > [email protected] > Subject: Re: [Help-glpk] [Fwd: I: Modelling binary variable] > > On Tue, 8 Oct 2013, Meketon, Marc wrote: > > > Are you sure that Z = Q[2]-Q[1]? For the case where x[1]=1, x[2]=0, > > x[3]=0, we have Q[1]=0, Q[2]=0, Q[3]=1, and then Q[3]-Q[2] = 1 which is the > > correct answer. > > Z = Q[1]-Q[2] > he has Q sorted in non-ascending order. > 00 no ones 0-0=0 > 10 one one 1-0=1 > 11 two or more ones 1-1=0 > exactly what you want > The size of N does not matter. > > Meketon's code has the advantage of ease of coding and understanding, but > it doubles the dimensionality. > > Assume one has an integer expression sum: > 0<=sum<=N > One wants z==1 iff sum==1 else 0 > Define N2lo=floor(N/2), N2hi=ceil(N/2) > Note N=N2lo+N2hi, H2hi-N2lo in {0, 1} > Add two (not N) more integer variables: > 0<=a<=N2hi > b binary > > require > sum=2a+b > z>=b-a > z<=b > z<=(1-N2hi/N2lo)b - a/N2lo + N2hi/N2lo > > The last constraint on z should be multiplied by N2lo to ensure > integrality of the coefficients. > > Done. > > The first two constraints on z are fairly obvious. > The last needs more explanation. > > The diagram below is for N==7. > > 3 0 - > 2 0 0 > a 1 0 0 > 0 0 1 > > 0 1 > b > > The rectangle gives the values of z for all valid combinations of a and > b. > The given constraints on z are all facets of the polyhedron. > The first is for facet (0, 0, 0)(0, 1, 1)(1, 0, 0). > The second for facet (0, 0, 0)(0, 1, 1)(N2hi, 0, 0). > The third for facet (N2lo, 1, 0)(0, 1, 1)(N2hi, 0, 0). > Substitution will verify. > > > Note that exhaustive testing is possible: > The number of combinations that need testing is at most 2*(N+1)**2 . > > -- > Michael [email protected] > "On Monday, I'm gonna have to tell my kindergarten class, whom I teach > not to run with scissors, that my fiance ran me through with a > broadsword." -- Lily > > This e-mail and any attachments may be confidential or legally > privileged. If you received this message in error or are not the intended > recipient, you should destroy the e-mail message and any attachments or > copies, and you are prohibited from retaining, distributing, disclosing > or using any information contained herein. Please inform us of the > erroneous delivery by return e-mail. Thank you for your cooperation. -- http://www.fastmail.fm - Send your email first class _______________________________________________ Help-glpk mailing list [email protected] https://lists.gnu.org/mailman/listinfo/help-glpk
