Semidefinite programming
Awaiting user inputI 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 opensource solver SCIP. This is also for mixed integers. Since there are no mixed integer constraints in my problem, are there any faster solvers?

The SDP plugin should be able to solve non mixedinteger 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 cuttingplane approach can be quite beneficial in certain scenarios. This method involves iteratively formulating MixedInteger 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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