Linear, Cone and Semidefinite Programming

Some Applications of Polynomial Optimization in Operations Research and Real-Time Decision Making

  • Amir Ali Ahmadi
  • Anirudha Majumdar
    • Categories Convex Optimization, Global Optimization, Linear, Cone and Semidefinite Programming Tags , , , ,

    We demonstrate applications of algebraic techniques that optimize and certify polynomial inequalities to problems of interest in the operations research and transportation engineering communities. Three problems are considered: (i) wireless coverage of targeted geographical regions with guaranteed signal quality and minimum transmission power, (ii) computing real-time certificates of collision avoidance for a simple model of … Read more

    Quantum and classical coin-flipping protocols based on bit-commitment and their point games

  • Ashwin Nayak
  • Jamie Sikora
  • Levent Tunçel
    • Categories Linear Programming, Second-Order Cone Programming, Semi-definite Programming Tags , , , ,

    We focus on a family of quantum coin-flipping protocols based on quantum bit-commitment. We discuss how the semidefinite programming formulations of cheating strategies can be reduced to optimizing a linear combination of fidelity functions over a polytope. These turn out to be much simpler semidefinite programs which can be modelled using second-order cone programming problems. … Read more

    Extended Formulations for Independence Polytopes of Regular Matroids

  • Volker Kaibel
  • Jon Lee
  • Matthias Walter
  • Stefan Weltge
    • Categories Linear Programming, Polyhedra Tags , , ,

    We show that the independence polytope of every regular matroid has an extended formulation of size quadratic in the size of its ground set. This generalizes a similar statement for (co-)graphic matroids, which is a simple consequence of Martin’s extended formulation for the spanning-tree polytope. In our construction, we make use of Seymour’s decomposition theorem … Read more

    Lower Bounds on Complexity of Lyapunov Functions for Switched Linear Systems

  • Amir Ali Ahmadi
  • Raphaël Jungers
    • Categories Convex Optimization, Semi-definite Programming, Systems governed by Differential Equations Optimization Tags , , ,

    We show that for any positive integer $d$, there are families of switched linear systems—in fixed dimension and defined by two matrices only—that are stable under arbitrary switching but do not admit (i) a polynomial Lyapunov function of degree $\leq d$, or (ii) a polytopic Lyapunov function with $\leq d$ facets, or (iii) a piecewise … Read more

    New bounds for the max-hBccut and chromatic number of a graph

  • Renata Sotirov
  • Edwin van Dam
    • Categories Combinatorial Optimization, Semi-definite Programming Tags , , , , , , ,

    We consider several semidefinite programming relaxations for the max-$k$-cut problem, with increasing complexity. The optimal solution of the weakest presented semidefinite programming relaxation has a closed form expression that includes the largest Laplacian eigenvalue of the graph under consideration. This is the first known eigenvalue bound for the max-$k$-cut when $k>2$ that is applicable to … Read more

    A strong polynomial gradient algorithm in Linear Programming

  • Peter A. Bruijs
    • Categories Convex Optimization, Linear Programming

    It has been an open question whether the Linear Programming (LP) problem can be solved in strong polynomial time. The simplex algorithm does not offer a polynomial bound, and polynomial algorithms by Khachiyan and Karmarkar don’t have the strong characteristic. The curious fact that non-linear algorithms would be needed to deliver the strongest complexity result … Read more

    A Framework for Applying Subgradient Methods to Conic Optimization Problems (version 2)

  • James Renegar
    • Categories Convex Optimization, Linear, Cone and Semidefinite Programming Tags ,

    A framework is presented whereby a general convex conic optimization problem is transformed into an equivalent convex optimization problem whose only constraints are linear equations and whose objective function is Lipschitz continuous. Virtually any subgradient method can be applied to solve the equivalent problem. Two methods are analyzed. (In version 2, the development of algorithms … Read more

    Extension and Implementation of Homogeneous Self-dual Methods for Symmetric Cones under Uncertainty

  • Baha Alzalg
  • Francesca Maggioni
  • Sebastiano Vitali
    • Categories Linear, Cone and Semidefinite Programming, Stochastic Programming

    Homogeneous self-dual algorithms for stochastic semidefinite programs with finite event space has been proposed by Jin et al. in [12]. Alzalg [8], has adopted their work to derive homogeneous self-dual algorithms for stochastic second-order programs with finite event space. In this paper, we generalize these two results to derive homogeneous self-dual algorithms for stochastic programs … Read more

    Parallelizing the dual revised simplex method

  • J. A. Julian Hall
  • Q. Huangfu
    • Categories Linear Programming, Parallel Algorithms

    This paper introduces the design and implementation of two parallel dual simplex solvers for general large scale sparse linear programming problems. One approach, called PAMI, extends a relatively unknown pivoting strategy called suboptimization and exploits parallelism across multiple iterations. The other, called SIP, exploits purely single iteration parallelism by overlapping computational components when possible. Computational … Read more