Jie Wang

Strengthening Lasserre’s Hierarchy in Real and Complex Polynomial Optimization

  • Jie Wang
    • Categories Constrained Nonlinear Optimization, Global Optimization Theory, Semi-definite Programming Tags , , , , ,

    We establish a connection between multiplication operators and shift operators. Moreover, we derive positive semidefinite conditions of finite rank moment sequences and use these conditions to strengthen Lasserre’s hierarchy for real and complex polynomial optimization. Integration of the strengthening technique with sparsity is considered. Extensive numerical experiments show that our strengthening technique can significantly improve … Read more

    Sparse Polynomial Optimization with Unbounded Sets

  • Jie Wang
    • Categories Constrained Nonlinear Optimization, Global Optimization Theory, Semi-definite Programming Tags , , , ,

    This paper considers sparse polynomial optimization with unbounded sets. When the problem possesses correlative sparsity, we propose a sparse homogenized Moment-SOS hierarchy with perturbations to solve it. The new hierarchy introduces one extra auxiliary variable for each variable clique according to the correlative sparsity pattern. Under the running intersection property, we prove that this hierarchy … Read more

    A real moment-HSOS hierarchy for complex polynomial optimization with real coefficients

  • Jie Wang
  • Victor Magron
    • Categories Global Optimization, Global Optimization Theory, Semi-definite Programming Tags , , , ,

    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

    A more efficient reformulation of complex SDP as real SDP

  • Jie Wang
    • Categories Global Optimization Theory, Semi-definite Programming Tags , , , ,

    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

    A Moment-SOS Hierarchy for Robust Polynomial Matrix Inequality Optimization with SOS-Convexity

  • Jie Wang
  • Feng Guo
    • Categories Global Optimization Applications, Robust Optimization, Semi-definite Programming Tags , , , ,

    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

    Solving low-rank semidefinite programs via manifold optimization

  • Jie Wang
    • Categories Constrained Nonlinear Optimization, Optimization Software Benchmark, Semi-definite Programming Tags , , , , ,

    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

    Weighted Geometric Mean, Minimum Mediated Set, and Optimal Second-Order Cone Representation

  • Jie Wang
    • Categories Cone Programming, Convex Optimization, Second-Order Cone Programming Tags , , , ,

    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

    Certifying Global Optimality of AC-OPF Solutions via sparse polynomial optimization

  • Jie Wang
  • Victor Magron
  • Jean-Bernard Lasserre
    • Categories Global Optimization Applications, Semi-definite Programming Tags , , , , ,

    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