Linear, Cone and Semidefinite Programming

On the Convergence of the Entropy-Exponential Penalty Trajectories and Generalized Proximal Point Methods in Semidefinite Optimization

  • Orizon Pereira Ferreira
  • Paulo Roberto Oliveira
  • R. C. M. Silva
    • Categories Linear, Cone and Semidefinite Programming, Semi-definite Programming Tags , , ,

    The convergence of primal and dual central paths associated to entropy and exponential functions, respectively, for semidefinite programming problem are studied in this paper. As an application, the proximal point method with the Kullback-Leibler distance applied to semidefinite programming problems is considered, and the convergence of primal and dual sequences is proved. Citation Journal of … Read more

    Sum of Squares Method for Sensor Network Localization

  • Jiawang Nie
    • Categories Combinatorial Optimization, Linear, Cone and Semidefinite Programming, Network Optimization Tags , , , , , ,

    We formulate the sensor network localization problem as finding the global minimizer of a quartic polynomial. Then sum of squares (SOS) relaxations can be applied to solve it. However, the general SOS relaxations are too expensive to implement for large problems. Exploiting the special features of this polynomial, we propose a new structured SOS relaxation, … Read more

    An Extension of a Minimax Approach to Multiple Classification

  • Tomonari Kitahara
  • Shinji Mizuno
  • Kazuhide Nakata
    • Categories Data-Mining, Second-Order Cone Programming Tags , ,

    When the mean vectors and the covariance matrices of two classes are available in a binary classification problem, Lanckriet et al.\ \cite{mpm} propose a minimax approach for finding a linear classifier which minimizes the worst-case (maximum) misclassification probability. We extend the minimax approach to a multiple classification problem, where the number $m$ of classes could … Read more

    Identifying Redundant Linear Constraints in Systems of Linear Matrix Inequality Constraints

  • Shafiu Jibrin
  • Daniel Stover
    • Categories Semi-definite Programming Tags , , ,

    Semidefinite programming has been an interesting and active area of research for several years. In semidefinite programming one optimizes a convex (often linear) objective function subject to a system of linear matrix inequality constraints. Despite its numerous applications, algorithms for solving semidefinite programming problems are restricted to problems of moderate size because the computation time … Read more

    Copositive programming motivated bounds on the stability and the chromatic number

  • Igor Dukanovic
  • Franz Rendl
    • Categories Graphs and Matroids, Linear, Cone and Semidefinite Programming Tags , , ,

    The Lovasz theta number of a graph G can be viewed as a semidefinite programming relaxation of the stability number of G. It has recently been shown that a copositive strengthening of this semidefinite program in fact equals the stability number of G. We introduce a related strengthening of the Lovasz theta number toward the … Read more

    An inexact primal-dual path following algorithm for convex quadratic SDP

  • Kim-Chuan Toh
    • Categories Linear, Cone and Semidefinite Programming Tags , , , ,

    We propose primal-dual path-following Mehrotra-type predictor-corrector methods for solving convex quadratic semidefinite programming (QSDP) problems of the form: $\min_{X} \{ \frac{1}{2} X\bullet {\cal Q}(X) + C\bullet X : {\cal A} (X) = b, X\succeq 0\}$, where ${\cal Q}$ is a self-adjoint positive semidefinite linear operator on ${\cal S}^n$, $b\in R^m$, and ${\cal A}$ is a … Read more

    Improved bounds for the symmetric rendezvous search problem on the line

  • Donglei Du
  • Q. Han
  • Juan Vera
  • Luis F. Zuluaga
    • Categories Combinatorial Optimization, Game Theory, Semi-definite Programming Tags , , ,

    A notorious open problem in the field of rendezvous search is to decide the rendezvous value of the symmetric rendezvous search problem on the line, when the initial distance apart between the two players is 2. We show that the symmetric rendezvous value is within the interval (4.1520, 4.2574), which considerably improves the previous best … Read more

    Robust Semidefinite Programming Approaches for Sensor Network Localization with Anchors

  • Nathan Krislock
  • Veronica Piccialli
  • Henry Wolkowicz
    • Categories Applications - OR and Management Sciences, Civil and Environmental Engineering, Linear, Cone and Semidefinite Programming Tags , , , , ,

    We derive a robust primal-dual interior-point algorithm for a semidefinite programming, SDP, relaxation for sensor localization with anchors and with noisy distance information. The relaxation is based on finding a Euclidean Distance Matrix, EDM, that is nearest in the Frobenius norm for the known noisy distances and that satisfies given upper and lower bounds on … Read more

    Target following algorithms for semidefinite programming

  • Chek Beng Chua
    • Categories Convex Optimization, Semi-definite Programming

    We present a target-following framework for semidefinite programming, which generalizes the target-following framework for linear programming. We use this framework to build weighted path-following interior-point algorithms of three distinct flavors: short-step, predictor-corrector, and large-update. These algorithms have worse-case iteration bounds that parallel their counterparts in linear programming. We further consider the problem of finding analytic … Read more

    Recognizing Underlying Sparsity in Optimization

  • Sunyoung Kim
  • Masakazu Kojima
  • Philippe L. Toint
    • Categories Combinatorial Optimization, Constrained Nonlinear Optimization, Semi-definite Programming Tags , , , ,

    Exploiting sparsity is essential to improve the efficiency of solving large optimization problems. We present a method for recognizing the underlying sparsity structure of a nonlinear partially separable problem, and show how the sparsity of the Hessian matrices of the problem’s functions can be improved by performing a nonsingular linear transformation in the space corresponding … Read more