Semidefinite programming
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I have an SDP problem (essentially it is an LP problem with two semidefinite matrices). Since Gurobi cannot solve it, please suggest an effective way to solve it. I saw the link SDP plugin of the open-source solver SCIP. This is also for mixed integers. Since there are no mixed integer constraints in my problem, are there any faster solvers?
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Gurobi Staff
The SDP plugin should be able to solve non mixed-integer SDPs as well, since it uses Mosek, DSDP, or SDPA as subsolver, which are able to solve continuous SDPs. Did you have a change to try it out?
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A cutting-plane approach can be quite beneficial in certain scenarios. This method involves iteratively formulating Mixed-Integer Linear Programs (MILPs) and solving them using tools like Gurobi.
While more advanced methods exist, a fundamental approach begins by initially disregarding the Semidefinite Programming (SDP) constraints imposed by matrix semidefiniteness. Instead, it solves the resulting MILP. If the solution satisfies the SDP constraints, it provides a valid solution, thereby completing the process. If not, constraints are added for each matrix \( A \) supposed to be positive definite, specifically requiring that \( \mathbf{v}^T A \mathbf{v} \geq 0 \), where \( \mathbf{v} \) is the eigenvector corresponding to the most negative eigenvalue of \( A \).
In general, it's important to note that this method does not guarantee obtaining a truly feasible solution, but each iteration typically produces a solution that is slightly less infeasible. However, it is advisable only when dedicated solvers specifically designed for solving mixed integer linear SDP problems are not available.
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