Comments (3)
Archenal,
Thanks for your interest in our work.
I'm not sure this is necessarily a bug. Factor width/structured subset approximations return an upper bound on the SDP objective. A dual infeasible objective means that the optimum objective value is -infinity. An upper bound of this may be -infinity (tight), or may be zero (returned by factor-width as you've observed). With a block-size of 5 or 10 (equivalent to the original problem), a dual infeasible certificate is returned. For any other block-size (like 1-3), a cost of zero is returned since b = 0 (and the returned x = 0). With a block-size of 4 the dual infeasible certificate is returned, not sure why that is the case.
Try it out by changing the BlockSize variable
load('Example_SDPfw.mat')
s = svd(full(A));
n_null = nnz(s <= 1e-8);
BlockSize = 4;
pars.fid = 0;
[x0, y0, info0] = sedumi(A, b, c, K, pars);
cost0 = c'*x0;
disp(info0)
% Same process through factorwidth
% opts=struct;
% opts.bfw = 1;
% opts.block = BlockSize;
%
% [Af, bf, cf, Kf] = factorwidth(A, b, c, K, opts);
% [xf, yf, infof] = sedumi(An, bn, cn, Kn, pars);
% costf = cf'*xf;
%disp(infof)
%Same process through decomposed_subset
[An, bn, cn, Kn, info_dec] = decomposed_subset(A, b, c, K, BlockSize);
[xn, yn, infon] = sedumi(An, bn, cn, Kn, pars);
xnr = decomposed_recover(xn, info_dec);
costn = c'*xnr;
disp(infon)
%eigenvalues of returned solution are in eX
count = 0;
X = cell(length(K.s), 1);
for i = 1:length(K.s)
Ksi = K.s(i);
xi = xnr(count + (1:Ksi^2));
X{i} = reshape(xi, Ksi, Ksi);
count = count + Ksi^2;
end
eX = cellfun(@eig, X, 'UniformOutput', false);
One issue with this problem is that A is rank deficient, and has two zero singular values. It looks like there is no avenue for facial reduction here, SieveSDP came up dry.
Hopefully this answers your question.
Best,
Jared
from sdpfw.
Hi Archenal,
Jared has given a detailed explanation. I didn't see an error in our code factorwidth.m either. The problem you have is dual infeasible but primal feasible. In fact, the primal problem has an unbounded primal objective, which is -infinity. So, when using the factor-width for inner approximation, it will return an upper bound for the primal problem, which can be -infinity or a bounded value ( it can be zero in some tests).
If you change the primal cost function a little bit, you may see the problem become both primal and dual feasible. Here is the code:
load Example_SDPfw
opts.bfw = 1;
opts.nop = 10;
c = rand(size(A,1),1); % change the cost function, won't change primal feasibility
[x,y,info] = sedumi(A,b,c,K);
[Anew, bnew, cnew, Knew, infofw] = factorwidth(A,b,c,K,opts);
% reformulating the SDP
[xn,yn,infon] = sedumi(Anew,bnew,cnew,Knew);
Let's know if you have further questions.
Best,
Yang
from sdpfw.
Hi Jared and Yang,
Thanks for your kind reply and the helpful explanations.
In fact, the problem I had should be primal infeasible, and I used YALMIP to convert it into sedumi format. Based on your reply, I realized that YALMIP automatically dualized my problem, that was why the results returned by the solver confused me. Now, I use YALMIP to convert my problem in the primal format, and factorwidth.m works perfectly.
Again, thank you for developing and providing these useful tools.
from sdpfw.
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