Semi-definite Programming

Perturbation analysis of a class of conic programming problems under Jacobian uniqueness conditions

  • Ziran Yin
  • Liwei Zhang
    • Categories Linear, Cone and Semidefinite Programming, Second-Order Cone Programming, Semi-definite Programming

    We consider the stability of a class of parameterized conic programming problems which are more general than the C2-smooth parameterization. We show that when the Karush-Kuhn-Tucker (KKT) condition, the constraint nondegeneracy condition, the strict complementary condition and the second order sucient condition (named as Jacobian uniqueness conditions here) are satis ed at a feasible point of … Read more

    Perturbation analysis of nonlinear semidefinite programming under Jacobian uniqueness conditions

  • Ziran Yin
  • Liwei Zhang
    • Categories Semi-definite Programming

    We consider the stability of a class of parameterized nonlinear semidefinite programming problems whose objective function and constraint mapping all have second partial derivatives only with respect to the decision variable which are jointly continuous. We show that when the Karush-Kuhn-Tucker (KKT) condition, the constraint nondegeneracy condition, the strict complementary condition and the second order … Read more

    Exploiting sparsity for the min k-partition problem

  • Hassan L. Hijazi
  • Guanglei Wang
    • Categories 0-1 Programming, Semi-definite Programming Tags , , ,

    The minimum k-partition problem is a challenging combinatorial problem with a diverse set of applications ranging from telecommunications to sports scheduling. It generalizes the max-cut problem and has been extensively studied since the late sixties. Strong integer formulations proposed in the literature suffer from a prohibitive number of valid inequalities and integer variables. In this … Read more

    Lower bounds on matrix factorization ranks via noncommutative polynomial optimization

  • David de Laat
  • Monique Laurent
  • Sander Gribling
    • Categories Semi-definite Programming Tags ,

    We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the nonnegative rank, the positive semidefinite rank, and their symmetric analogues: the completely positive rank and the completely positive semidefinite rank. We study the convergence properties of our hierarchies, compare them … Read more

    Bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization

  • David de Laat
  • Monique Laurent
  • Sander Gribling
    • Categories Semi-definite Programming Tags , ,

    In this paper we study bipartite quantum correlations using techniques from tracial polynomial optimization. We construct a hierarchy of semidefinite programming lower bounds on the minimal entanglement dimension of a bipartite correlation. This hierarchy converges to a new parameter: the minimal average entanglement dimension, which measures the amount of entanglement needed to reproduce a quantum … Read more

    A Data-Driven Distributionally Robust Bound on the Expected Optimal Value of Uncertain Mixed 0-1 Linear Programming

  • Samuel Burer
  • Guanglin Xu
    • Categories Robust Optimization, Semi-definite Programming, Stochastic Programming

    This paper studies the expected optimal value of a mixed 0-1 programming problem with uncertain objective coefficients following a joint distribution. We assume that the true distribution is not known exactly, but a set of independent samples can be observed. Using the Wasserstein metric, we construct an ambiguity set centered at the empirical distribution from … Read more

    On Solving the Quadratic Shortest Path Problem

  • Hao Hu
  • Renata Sotirov
    • Categories Semi-definite Programming Tags , , ,

    The quadratic shortest path problem is the problem of finding a path in a directed graph such that the sum of interaction costs over all pairs of arcs on the path is minimized. We derive several semidefinite programming relaxations for the quadratic shortest path problem with a matrix variable of order $m+1$, where $m$ is … Read more

    A rounding procedure for semidefinite optimization

  • Ali Mohammad-Nezhad
  • Tamás Terlaky
    • Categories Semi-definite Programming Tags , , ,

    Recently, Mohammad-Nezhad and Terlaky studied the identification of the optimal partition for semidefinite optimization. An approximation of the optimal partition was obtained from a bounded sequence of solutions on, or in a neighborhood of the central path. Here, we use the approximation of the optimal partition in a rounding procedure to generate an approximate maximally … Read more

    Semidefinite Programming and Nash Equilibria in Bimatrix Games

  • Amir Ali Ahmadi
  • Jeffrey Zhang
    • Categories Game Theory, Semi-definite Programming Tags , ,

    We explore the power of semidefinite programming (SDP) for finding additive epsilon-approximate Nash equilibria in bimatrix games. We introduce an SDP relaxation for a quadratic programming formulation of the Nash equilibrium (NE) problem and provide a number of valid inequalities to improve the quality of the relaxation. If a rank-1 solution to this SDP is … Read more

    SDP-based Branch-and-Bound for Non-convex Quadratic Integer Optimization

  • Christoph Buchheim
  • Maribel Montenegro
  • Angelika Wiegele
    • Categories Quadratic Programming, Semi-definite Programming Tags , ,

    Semidefinite programming (SDP) relaxations have been intensively used for solving discrete quadratic optimization problems, in particular in the binary case. For the general non-convex integer case with box constraints, the branch-and-bound algorithm Q-MIST has been proposed [11], which is based on an extension of the well-known SDP-relaxation for max-cut. For solving the resulting SDPs, Q-MIST … Read more