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A branch-and-cut algorithm for solving mixed-integer semidefinite optimization problems. (English) Zbl 1440.90031
Summary: We consider a cutting-plane algorithm for solving mixed-integer semidefinite optimization (MISDO) problems. In this algorithm, the positive semidefinite (psd) constraint is relaxed, and the resultant mixed-integer linear optimization problem is solved repeatedly, imposing at each iteration a valid inequality for the psd constraint. We prove the convergence properties of the algorithm. Moreover, to speed up the computation, we devise a branch-and-cut algorithm, in which valid inequalities are dynamically added during a branch-and-bound procedure. We test the computational performance of our cutting-plane and branch-and-cut algorithms for three types of MISDO problem: random instances, computing restricted isometry constants, and robust truss topology design. Our experimental results demonstrate that, for many problem instances, our branch-and-cut algorithm delivered superior performance compared with general-purpose MISDO solvers in terms of computational efficiency and stability.
##### MSC:
 90C11 Mixed integer programming 90C22 Semidefinite programming 90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut
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