Linear, Cone and Semidefinite Programming

A Simple Variant of the Mizuno-Todd-Ye Predictor-Corrector Algorithm and its Objective-Function-Free Complexity

  • Tomonari Kitahara
  • Takashi Tsuchiya
    • Categories Linear Programming Tags , , , ,

    In this paper, we propose a simple variant of the Mizuno-Todd-Ye predictor-corrector algorithm for linear programming problem (LP). Our variant executes a natural finite termination procedure at each iteration and it is easy to implement the algorithm. Our algorithm admits an objective-function free polynomial-time complexity when it is applied to LPs whose dual feasible region … Read more

    Inner approximations for polynomial matrix inequalities and robust stability regions

  • Didier Henrion
  • Jean-Bernard Lasserre
    • Categories Robust Optimization, Semi-definite Programming Tags , , , , ,

    Following a polynomial approach, many robust fixed-order controller design problems can be formulated as optimization problems whose set of feasible solutions is modelled by parametrized polynomial matrix inequalities (PMI). These feasibility sets are typically nonconvex. Given a parametrized PMI set, we provide a hierarchy of linear matrix inequality (LMI) problems whose optimal solutions generate inner … Read more

    The Second Order Directional Derivative of Symmetric Matrix-valued Functions

  • Xiantao Xiao
  • Liwei Zhang
  • Ning Zhang
    • Categories Linear, Cone and Semidefinite Programming

    This paper focuses on the study of the second-order directional derivative of a symmetric matrix-valued function of the form $F(X)=P\mbox{diag}[f(\lambda_1(X)),\cdots,f(\lambda_n(X))]P^T$. For this purpose, we first adopt a direct way to derive the formula for the second-order directional derivative of any eigenvalue of a matrix in Torki \cite{Tor01}; Second, we establish a formula for the (parabolic) … Read more

    High accuracy solution of large scale semidefinite programs

  • Thomas Davi
  • Florian Jarre
    • Categories Semi-definite Programming Tags , ,

    We present a first order approach for solving semidefinite programs. Goal of this approach is to compute a solution of the SDP up to high accuracy in spite of using only partial second order information. We propose a hybrid approach that uses an accelerated projection method to generate an approximate solution and then switches to … Read more

    Solving large scale problems over the doubly nonnegative cone

  • Thomas Davi
  • Florian Jarre
    • Categories Semi-definite Programming Tags , , , ,

    The recent approach of solving large scale semidefinite programs with a first order method by minimizing an augmented primal-dual function is extended to doubly nonnegative programs. Regularity of the augmented primal-dual function is established under the condition of uniqueness and strict complementarity. The application to the doubly nonnegative cone is motivated by the fact that … Read more

    How bad is a gradient algorithm for linear programming?

  • Peter A. Bruijs
    • Categories Linear Programming, Linear, Cone and Semidefinite Programming

    In their 1972 paper ‘How good is the simplex algorithm ?’ Klee and Minty present a class of problems the simplex algorithm for linear programming (LP) is not able to solve in a polynomial way. Later developments have resulted in algorithms by Khachiyan and Karmarkar that do solve LP in a polynomial way, although the … Read more

    On Solving Biquadratic Optimization via Semidefinite Relaxation

  • Yuning Yang
  • Qingzhi Yang
    • Categories Semi-definite Programming Tags , , ,

    In this paper, we study a class of biquadratic optimization problems. We first relax the original problem to its semidefinite programming (SDP) problem and discuss the approximation ratio between them. Under some conditions, we show that the relaxed problem is tight. Then we consider how to approximately solve the problems in polynomial time. Under several … Read more

    Level methods uniformly optimal for composite and structured nonsmooth convex optimization

  • Guanghui Lan
    • Categories Convex and Nonsmooth Optimization, Linear, Cone and Semidefinite Programming, Stochastic Programming Tags , , , , ,

    The main goal of this paper is to develop uniformly optimal first-order methods for large-scale convex programming (CP). By uniform optimality we mean that the first-order methods themselves do not require the input of any problem parameters, but can still achieve the best possible iteration complexity bounds. To this end, we provide a substantial generalization … Read more

    Level methods uniformly optimal for composite and structured nonsmooth convex optimization

  • Guanghui Lan
    • Categories Convex and Nonsmooth Optimization, Linear, Cone and Semidefinite Programming, Stochastic Programming Tags , , , , ,

    The main goal of this paper is to develop uniformly optimal first-order methods for large-scale convex programming (CP). By uniform optimality we mean that the first-order methods themselves do not require the input of any problem parameters, but can still achieve the best possible iteration complexity bounds. To this end, we provide a substantial generalization … Read more

    Approximation algorithms for trilinear optimization with nonconvex constraints and its extensions

  • Yuning Yang
  • Qingzhi Yang
    • Categories Semi-definite Programming Tags , , , ,

    In this paper, we study trilinear optimization problems with nonconvex constraints under some assumptions. We first consider the semidefinite relaxation (SDR) of the original problem. Then motivated by So \cite{So2010}, we reduce the problem to that of determining the $L_2$-diameters of certain convex bodies, which can be approximately solved in deterministic polynomial-time. After the relaxed … Read more