Jie Wang This paper proposes a real moment-HSOS hierarchy for complex polynomial optimization problems with real coefficients. We show that this hierarchy provides the same sequence of lower bounds as the complex analogue, yet is much cheaper to solve. In addition, we prove that global optimality is achieved when the ranks of the moment matrix and certain … Read more
This note proposes a novel reformulation of complex semidefinite programs (SDPs) as real SDPs. As an application, we present an economical reformulation of complex SDP relaxations of complex polynomial optimization problems as real SDPs and derive some further reductions by exploiting structure of the complex SDP relaxations. Various numerical examples demonstrate that our new reformulation … Read more
We study a class of polynomial optimization problems with a robust polynomial matrix inequality constraint for which the uncertainty set is defined also by a polynomial matrix inequality (including robust polynomial semidefinite programs as a special case). Under certain SOS-convexity assumptions, we construct a hierarchy of moment-SOS relaxations for this problem to obtain convergent upper … Read more
We propose a manifold optimization approach to solve linear semidefinite programs (SDP) with low-rank solutions. This approach incorporates the augmented Lagrangian method and the Burer-Monteiro factorization, and features the adaptive strategies for updating the factorization size and the penalty parameter. We prove that the present algorithm can solve SDPs to global optimality, despite of the … Read more
We study optimal second-order cone representations for weighted geometric means, which turns out to be closely related to minimum mediated sets. Several lower bounds and upper bounds on the size of optimal second-order cone representations are proved. In the case of bivariate weighted geometric means (equivalently, one dimensional mediated sets), we are able to prove … Read more
We report the experimental results on certifying 1% global optimality of solutions of AC-OPF instances from PGLiB via the CS-TSSOS hierarchy — a moment-SOS based hierarchy that exploits both correlative and term sparsity, which can provide tighter SDP relaxations than Shor’s relaxation. Our numerical experiments demonstrate that the CS-TSSOS hierarchy scales well with the problem … Read more