Linear, Cone and Semidefinite Programming

Representation of distributionally robust chance-constraints

  • Jean-Bernard Lasserre
  • Tillmann Weisser
    • Categories Constrained Nonlinear Optimization, Semi-definite Programming, Stochastic Programming

    Given $X\subset R^n$, $\varepsilon \in (0,1)$, a parametrized family of probability distributions $(\mu_{a})_{a\in A}$ on $\Omega\subset R^p$, we consider the feasible set $X^*_\varepsilon\subset X$ associated with the {\em distributionally robust} chance-constraint \[X^*_\varepsilon\,=\,\{x\in X:\:{\rm Prob}_\mu[f(x,\omega)\,>\,0]> 1-\varepsilon,\,\forall\mu\in\mathscr{M}_a\},\] where $\mathscr{M}_a$ is the set of all possibles mixtures of distributions $\mu_a$, $a\in A$. For instance and typically, the family … Read more

    Moments and convex optimization for analysis and control of nonlinear partial differential equations

  • Didier Henrion
  • Jean-Bernard Lasserre
  • Milan Korda
    • Categories Distributed Control, Optimization of Systems modeled by PDEs, Semi-definite Programming Tags , , , ,

    This work presents a convex-optimization-based framework for analysis and control of nonlinear partial differential equations. The approach uses a particular weak embedding of the nonlinear PDE, resulting in a \emph{linear} equation in the space of Borel measures. This equation is then used as a constraint of an infinite-dimensional linear programming problem (LP). This LP is … Read more

    Solving Pooling Problems by LP and SOCP Relaxations and Rescheduling Methods

  • Sunyoung Kim
  • Masaki Kimizuka
  • Makoto Yamashita
    • Categories Applications - Science and Engineering, Linear, Cone and Semidefinite Programming Tags , , , , ,

    The pooling problem is an important industrial problem in the class of network flow problems for allocating gas flow in pipeline transportation networks. For P-formulation of the pooling problem with time discretization, we propose second order cone programming (SOCP) and linear programming (LP) relaxations and prove that they obtain the same optimal value as the … Read more

    Improving the linear relaxation of maximum hBccut with semidefinite-based constraints

  • Miguel F. Anjos
  • SĂ©bastien Le Digabel
  • Vilmar JeftĂ© Rodrigues de Sousa
    • Categories Combinatorial Optimization, Linear, Cone and Semidefinite Programming Tags , , , ,

    We consider the maximum $k$-cut problem that involves partitioning the vertex set of a graph into $k$ subsets such that the sum of the weights of the edges joining vertices in different subsets is maximized. The associated semidefinite programming (SDP) relaxation is known to provide strong bounds, but it has a high computational cost. We … Read more

    Using Nemirovski’s Mirror-Prox method as Basic Procedure in Chubanov’s method for solving homogeneous feasibility problems

  • Cornelis Roos
  • Wei Zhang
    • Categories Applications - OR and Management Sciences, Linear Programming Tags ,

    We introduce a new variant of Chubanov’s method for solving linear homogeneous systems with positive variables. In the \BP\ we use a recently introduced cut in combination with Nemirovski’s Mirror-Prox method. We show that the cut requires at most $O(n^3)$ time, just as Chabonov’s cut. In an earlier paper it was shown that the new … Read more

    BBCPOP: A Sparse Doubly Nonnegative Relaxation of Polynomial Optimization Problems with Binary, Box and Complementarity Constraints

  • Naoki Ito
  • Sunyoung Kim
  • Masakazu Kojima
  • Kim-Chuan Toh
  • Akiko Takeda
    • Categories Convex Optimization, Global Optimization, Linear, Cone and Semidefinite Programming Tags , , , , , , ,

    The software package BBCPOP is a MATLAB implementation of a hierarchy of sparse doubly nonnegative (DNN) relaxations of a class of polynomial optimization (minimization) problems (POPs) with binary, box and complementarity (BBC) constraints. Given a POP in the class and a relaxation order, BBCPOP constructs a simple conic optimization problem (COP), which serves as a … Read more

    User Manual for BBCPOP: A Sparse Doubly Nonnegative Relaxation of Polynomial Optimization Problems with Binary, Box and Complementarity Constraints

  • Naoki Ito
  • Sunyoung Kim
  • Masakazu Kojima
  • Kim-Chuan Toh
  • Akiko Takeda
    • Categories Convex Optimization, Global Optimization, Linear, Cone and Semidefinite Programming Tags , , , ,

    BBCPOP proposed in [4] is a MATLAB implementation of a hierarchy of sparse doubly nonnegative (DNN) relaxations of a class of polynomial optimization (minimization) problems (POPs) with binary, box and complementarity constraints. Given a POP in the class and a relaxation order (or a hierarchy level), BBCPOP constructs a simple conic optimization problem (COP), which … Read more

    Convex optimization under combinatorial sparsity constraints

  • Christoph Buchheim
  • Emiliano Traversi
    • Categories Graphs and Matroids, Linear, Cone and Semidefinite Programming Tags , , ,

    We present a heuristic approach for convex optimization problems containing sparsity constraints. The latter can be cardinality constraints, but our approach also covers more complex constraints on the support of the solution. For the special case that the support is required to belong to a matroid, we propose an exchange heuristic adapting the support in … Read more

    Entropic proximal operators for nonnegative trigonometric polynomials

  • Hsiao-Han Chao
  • Lieven Vandenberghe
    • Categories Convex and Nonsmooth Optimization, Linear, Cone and Semidefinite Programming

    Signal processing applications of semidefinite optimization are often rooted in sum-of-squares representations of nonnegative trigonometric polynomials. Interior-point solvers for semidefinite optimization can handle constraints of this form with a per-iteration-complexity that is cubic in the degree of the trigonometric polynomial. The purpose of this paper is to discuss first-order methods with a lower complexity per … Read more

    Hadamard Directional Diff erentiability of the Optimal Value of a Linear Second-order Conic Programming Problem

  • Duan Qingsong
  • Liwei Zhang
  • Sainan Zhang
    • Categories Second-Order Cone Programming, Stochastic Programming Tags , , ,

    In this paper, we consider perturbation properties of a linear second-order conic optimization problem and its Lagrange dual in which all parameters in the problem are perturbed. We prove the upper semi-continuity of solution mappings for the primal problem and the Lagrange dual problem. We demonstrate that the optimal value function can be expressed as … Read more