Linear, Cone and Semidefinite Programming

Polynomial worst-case iteration complexity of quasi-Newton primal-dual interior point algorithms for linear programming

  • Francisco N. C. Sobral
  • Jacek Gondzio
    • Categories Linear Programming, Linear, Cone and Semidefinite Programming, Nonlinear Optimization Tags , , ,

    Quasi-Newton methods are well known techniques for large-scale numerical optimization. They use an approximation of the Hessian in optimization problems or the Jacobian in system of nonlinear equations. In the Interior Point context, quasi-Newton algorithms compute low-rank updates of the matrix associated with the Newton systems, instead of computing it from scratch at every iteration. … Read more

    Production Theory for Constrained Linear Activity Models

  • Somdeb Lahiri
    • Categories Finance and Economics, Linear Programming, Production and Logistics Tags , ,

    The purpose of this paper is to generalize the framework of activity analysis discussed in Villar (2003) and obtain similar results concerning solvability. We generalize the model due to Villar (2003), without requiring any dimensional requirements on the activity matrices and by introducing a model of activity analysis in which each activity may (or may … 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

    Superadditive duality and convex hulls for mixed-integer conic optimization

  • Akshay Gupte
  • Temitayo Ajayi
  • Amin Khademi
  • Andrew J. Schaefer
    • Categories (Mixed) Integer Nonlinear Programming, Cone Programming, Cutting Plane Approaches Tags , , ,

    We present an infinite family of linear valid inequalities for a mixed-integer conic program, and prove that these inequalities describe the convex hull of the feasible set when this set is bounded and described by integral data. The main element of our proof is to establish a new strong superadditive dual for mixed-integer conic programming … Read more

    A primal-dual majorization-minimization method for large-scale linear programs

  • Xin-Wei Liu
  • Yu-Hong Dai
  • Yakui Huang
    • Categories Linear Programming Tags , , , ,

    We present a primal-dual majorization-minimization method for solving large-scale linear programs. A smooth barrier augmented Lagrangian (SBAL) function with strict convexity for the dual linear program is derived. The majorization-minimization approach is naturally introduced to develop the smoothness and convexity of the SBAL function. Our method only depends on a factorization of the constant matrix … Read more

    A Strengthened SDP Relaxation for Quadratic Optimization Over the Stiefel Manifold

  • Samuel Burer
  • Kyungchan Park
    • Categories Global Optimization, Quadratic Programming, Semi-definite Programming Tags , ,

    We study semidefinite programming (SDP) relaxations for the NP-hard problem of globally optimizing a quadratic function over the Stiefel manifold. We introduce a strengthened relaxation based on two recent ideas in the literature: (i) a tailored SDP for objectives with a block-diagonal Hessian; (ii) and the use of the Kronecker matrix product to construct SDP relaxations. Using synthetic instances on … Read more

    A copositive framework for analysis of hybrid Ising-classical algorithms

  • David E. Bernal Neira
  • Robin Brown
    • Categories (Mixed) Integer Nonlinear Programming, Cone Programming, Cutting Plane Approaches Tags , , , , ,

    Recent years have seen significant advances in quantum/quantum-inspired technologies capable of approximately searching for the ground state of Ising spin Hamiltonians. The promise of leveraging such technologies to accelerate the solution of difficult optimization problems has spurred an increased interest in exploring methods to integrate Ising problems as part of their solution process, with existing … Read more

    Automorphisms of rank-one generated hyperbolicity cones and their derivative relaxations

  • Masaru Ito
  • Bruno F. Lourenço
    • Categories Cone Programming, Linear, Cone and Semidefinite Programming Tags , ,

    A hyperbolicity cone is said to be rank-one generated (ROG) if all its extreme rays have rank one, where the rank is computed with respect the underlying hyperbolic polynomial. This is a natural class of hyperbolicity cones which are strictly more general than the ROG spectrahedral cones. In this work, we present a study of … 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

    A Newton-CG based barrier method for finding a second-order stationary point of nonconvex conic optimization with complexity guarantees

  • Chuan He
  • Zhaosong Lu
    • Categories Linear, Cone and Semidefinite Programming, Nonlinear Optimization Tags , , , , ,

    In this paper we consider finding an approximate second-order stationary point (SOSP) of nonconvex conic optimization that minimizes a twice differentiable function over the intersection of an affine subspace and a convex cone. In particular, we propose a Newton-conjugate gradient (Newton-CG) based barrier method for finding an $(\epsilon,\sqrt{\epsilon})$-SOSP of this problem. Our method is not … Read more