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Hello, I was wondering if there are algorithms for (linear) Semidefinite Programs (SDP) out there, that converge towards a maximally complementary solution, even if strict complementary does not hold. In particular: My SDP (for which I know stricly feasible starting points (primal and dual)) does not have any stricty complementary optimal solutions. Because of that I am interested in at least maximally complementary solutions. If yes, I would like to find a proof. Thanks for any help.

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The standard approach is to embed the original SDP in a self dual formulation that has strictly feasible primal/dual solutions, solve the self dual formulation and then reach conclusions about the original problem from the solution of the self-dual problem.
See for example:

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.37.679&rep=rep1&type=pdf Infeasible-start semidefinite programming algorithms via self-dual embeddings

Unfortunately, I believe that this will only tell you when the problem has no strictly complementary primal-dual solutions and not necessarily give you a maximally complementary solution.

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  • $\begingroup$ Well they certainly don't tell you, wheter the path following method actually converges towards maximally complementary solutions or not. Of course when strict complementary holds, than it usually will (proved by Güler and Ye? ). But that might not be true in my case. Thank you for your reply though! $\endgroup$
    – Charles
    Jul 9 '12 at 17:58

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