Linear, Cone and Semidefinite Programming

Primal-Dual Interior-Point Algorithms for Semidefinite Optimization Based on a Simple Kernel Function

  • Y.Q. Bai
  • Cornelis Roos
  • G. Q. Wang
    • Categories Linear, Cone and Semidefinite Programming, Semi-definite Programming Tags , , , ,

    Interior-point methods (IPMs) for semidefinite optimization (SDO) have been studied intensively, due to their polynomial complexity and practical efficiency. Recently, J.Peng et al. introduced so-called self-regular kernel (and barrier) functions and designed primal-dual interior-point algorithms based on self-regular proximity for linear optimization (LO) problems. They have also extended the approach for LO to SDO. In … Read more

    Universal Duality in Conic Convex Optimization

  • Dianne P. O'Leary
  • Simon P. Schurr
  • André L. Tits
    • Categories Linear, Cone and Semidefinite Programming Tags , , , ,

    Given a primal-dual pair of linear programs, it is well known that if their optimal values are viewed as lying on the extended real line, then the duality gap is zero, unless both problems are infeasible, in which case the optimal values are +infinity and -infinity. In contrast, for optimization problems over nonpolyhedral convex cones, … Read more

    Pointillism via Linear Programming

  • Robert Bosch
  • Adrianne Herman
    • Categories Airline Optimization, Applications - Science and Engineering, Linear Programming Tags ,

    Pointillism is a painting technique in which the painter places dots of paint on the canvas in such a way that they blend together into desired forms when viewed from a distance. In this brief note, we describe how to use linear programming to construct a pointillist portrait. CitationDept. of Mathematics, Oberlin College, Oberlin, OH … Read more

    A semidefinite programming based polyhedral cut and price algorithm for the maxcut problem

  • Kartik Krishnan
  • John E. Mitchell
    • Categories Branch and Cut Algorithms, Semi-definite Programming Tags , , ,

    We investigate solution of the maximum cut problem using a polyhedral cut and price approach. The dual of the well-known SDP relaxation of maxcut is formulated as a semi-infinite linear programming problem, which is solved within an interior point cutting plane algorithm in a dual setting; this constitutes the pricing (column generation) phase of the … Read more

    Implementation of Infinite Dimensional Interior Point Method for Solving Multi-criteria Linear-Quadratic Control Problem

  • Leonid Faybusovich
  • Takashi Tsuchiya
  • Thanasak Mouktonglang
    • Categories Control Applications, Infinite Dimensional Optimization, Linear, Cone and Semidefinite Programming Tags , ,

    We describe an implementation of an infinite-dimensional primal-dual algorithm based on the Nesterov-Todd direction. Several applications to both continuous and discrete-time multi-criteria linear-quadratic control problems and linear-quadratic control problem with quadratic constraints are described. Numerical results show a very fast convergence (typically, within 3-4 iterations) to optimal solutions CitationPreprint, May, 2004, University of Notre DameArticleDownload … Read more

    Sensitivity analysis for linear optimization problem with fuzzy data in the objective function

  • Stephan Dempe
  • Tatiana Starostina
    • Categories Linear Programming, Robust Optimization Tags , ,

    Linear programming problems with fuzzy coefficients in the objective function are considered. Emphasis is on the dependence of the optimal solution from linear perturbations of the membership functions of the objective function coefficients as well as on the computation of a robust solution of the fuzzy linear problem if the membership functions are not surely … Read more

    An Iterative Solver-Based Infeasible Primal-Dual Path-Following Algorithm for Convex QP

  • Zhaosong Lu
  • Renato D.C. Monteiro
  • Jerome W. O'Neal
    • Categories Linear, Cone and Semidefinite Programming, Quadratic Programming Tags , , ,

    In this paper we develop an interior-point primal-dual long-step path-following algorithm for convex quadratic programming (CQP) whose search directions are computed by means of an iterative (linear system) solver. We propose a new linear system, which we refer to as the \emph{augmented normal equation} (ANE), to determine the primal-dual search directions. Since the condition number … Read more

    SDP vs. LP relaxations for the moment approach in some performance evaluation problems

  • Jean-Bernard Lasserre
  • Thomas Prieto-Rumeau
    • Categories Applications - OR and Management Sciences, Finance and Economics, Semi-definite Programming

    Given a Markov process we are interested in the numerical computation of the moments of the exit time from a bounded domain. We use a moment approach which, together with appropriate semidefinite positivity moment conditions, yields a sequence of semidefinite programs (or SDP relaxations), depending on the number of moments considered, that provide a sequence … Read more

    Cutting plane algorithms for robust conic convex optimization

  • Juan Alfredo Gomez
  • Walter Gómez
    • Categories Linear, Cone and Semidefinite Programming, Semi-infinite Programming Tags , ,

    In the paper we study some well-known cases of nonlinear programming problems, presenting them as instances of Inexact Linear Programming. The class of problems considered contains, in particular, semidefinite programming, second order cone programming and special cases of inexact semidefinite programming. Strong duality results for the nonlinear problems studied are obtained via the Lagrangian duality. … Read more

    An Extension of Sums of Squares Relaxations to Polynomial Optimization Problems over Symmetric Cones

  • Masakazu Kojima
  • Masakazu Muramatsu
    • Categories Global Optimization Theory, Semi-definite Programming Tags , , , , , ,

    This paper is based on a recent work by Kojima which extended sums of squares relaxations of polynomial optimization problems to polynomial semidefinite programs. Let ${\cal E}$ and ${\cal E}_+$ be a finite dimensional real vector space and a symmetric cone embedded in ${\cal E}$; examples of $\calE$ and $\calE_+$ include a pair of the … Read more