Doubly stochastic matrix in linear programming
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How may I get the vector x by using linprog(f,A,b), where b=Wy(y is a known vector) and W is all possible doubly stochastic matrix? Or other methods will work for lp given constraints involve doubly stochastic matrix, especially if W is high dimensional and enumeration seems infeasible?
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Torsten
2015-1-16
You mean how you can formulate the above problem for linprog ?
min: f'x
s.c.
A*x-Z*y=0
sum_i z_ij = 1
sum_j z_ij = 1
0 <= z_ij <= 1
Or what exactly are you asking for ?
Best wishes
Torsten.
更多回答（1 个）
This assumes that A will always be non-empty.
[m,n]=size(A);
p=m^2+n; %all unknowns
fwx=f; fwx(p)=0;
Awx=[kron(-y.',speye(m)), A];
bwx=zeros(m,1);
C= kron(speye(m), ones(1,m));
R= kron(ones(1,m), speye(m));
Aeq=[C;R]; Aeq(end,p)=0;
beq= ones(2*m,1);
lb=-inf(1,p); lb(1:m^2)=0;
ub=+inf(1,p; lb(1:m^2)=1;
WX=linprog(fwx,Awx,bwx,Aeq,beq,lb,ub);
W=reshape(WX(1:m^2),m,[]);
x=WX(m^2+1:p);
1 个评论
Matt J
2015-1-16
No, and actually just the opposite.
You mean you definitely want equality in
A*x-Z*y=0
If so, modify the call to linprog as follows
WX=linprog(fwx,[],[],[Aeq;Awx], [beq; bwx ],lb,ub);
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