linear matrix inequalities

A Framework for Solving Chance-Constrained Linear Matrix Inequality Programs

  • Jianqiang Cheng
  • Miguel A. Lejeune
  • Roya Karimi
    • Categories Stochastic Programming Tags , , , ,

    We propose a novel partial sample average approximation (PSAA) framework to solve the two main types of chance-constrained linear matrix inequality (CCLMI) problems: CCLMI with random technology matrix, and CCLMI with random right-hand side. We propose a series of computationally tractable PSAA-based approximations for CCLMI problems, analyze their properties, and derive sufficient conditions ensuring convexity. … Read more

    Computing Weighted Analytic Center for Linear Matrix Inequalities Using Infeasible Newton’s Method

  • Shafiu Jibrin
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Semi-definite Programming Tags , ,

    We study the problem of computing weighted analytic center for system of linear matrix inequality constraints. The problem can be solved using the Standard Newton’s method. However, this approach requires that a starting point in the interior point of the feasible region be given or a Phase I problem be solved. We address the problem … Read more

    SPECTRA – a Maple library for solving linear matrix inequalities in exact arithmetic

  • Didier Henrion
  • Simone Naldi
  • Mohab Safey El Din
    • Categories Semi-definite Programming Tags , , , ,

    This document briefly describes our freely distributed Maple library {\sc spectra}, for Semidefinite Programming solved Exactly with Computational Tools of Real Algebra. It solves linear matrix inequalities in exact arithmetic and it is targeted to small-size, possibly degenerate problems for which symbolic infeasibility or feasibility certificates are required. ArticleDownload View PDF

    Solving rank-constrained semidefinite programs in exact arithmetic

  • Simone Naldi
    • Categories Semi-definite Programming Tags , , , , , , , ,

    We consider the problem of minimizing a linear function over an affine section of the cone of positive semidefinite matrices, with the additional constraint that the feasible matrix has prescribed rank. When the rank constraint is active, this is a non-convex optimization problem, otherwise it is a semidefinite program. Both find numerous applications especially in … Read more

    Simple Approximations of Semialgebraic Sets and their Applications to Control

  • Fabrizio Dabbene
  • Didier Henrion
  • Constantino Lagoa
    • Categories Control Applications, Semi-definite Programming, Statistics Tags , , ,

    Many uncertainty sets encountered in control systems analysis and design can be expressed in terms of semialgebraic sets, that is as the intersection of sets described by means of polynomial inequalities. Important examples are for instance the solution set of linear matrix inequalities or the Schur/Hurwitz stability domains. These sets often have very complicated shapes … Read more

    A Randomized Cutting Plane Method with Probabilistic Geometric Convergence

  • Fabrizio Dabbene
  • Boris T. Polyak
  • Pavel S. Shcherbakov
    • Categories Convex Optimization, Nonsmooth Optimization Tags , , ,

    We propose a randomized method for general convex optimization problems; namely, the minimization of a linear function over a convex body. The idea is to generate N random points inside the body, choose the best one and cut the part of the body defined by the linear constraint. We first analyze the convergence properties of … Read more

    On Safe Tractable Approximations of Chance Constrained Linear Matrix Inequalities

  • Aharon Ben-Tal
  • Arkadi Nemirovski
    • Categories Linear, Cone and Semidefinite Programming, Robust Optimization, Stochastic Programming Tags , ,

    In the paper, we consider the chance constrained version $$ \Prob\{A_0[x]+\sum_{i=1}^d\zeta_i A_i[x]\succeq0\}\geq1-\epsilon, $$ of an affinely perturbed Linear Matrix Inequality constraint; here $A_i[x]$ are symmetric matrices affinely depending on the decision vector $x$, and $\zeta_1,…,\zeta_d$ are independent of each other random perturbations with light tail distributions (e.g., bounded or Gaussian). Constraints of this type, playing … 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

    Multivariate Nonnegative Quadratic Mappings

  • Zhi-Quan Luo
  • Jos F. Sturm
  • Shuzhong Zhang
    • Categories Linear, Cone and Semidefinite Programming Tags , ,

    In this paper we study several issues related to the characterization of specific classes of multivariate quadratic mappings that are nonnegative over a given domain, with nonnegativity defined by a pre-specified conic order. In particular, we consider the set (cone) of nonnegative quadratic mappings defined with respect to the positive semidefinite matrix cone, and study … Read more