Linear, Cone and Semidefinite Programming

Solving conic optimization problems via self-dual embedding and facial reduction: a unified approach

  • Erling D. Andersen
  • Henrik A. Friberg
  • Frank Permenter
    • Categories Linear, Cone and Semidefinite Programming Tags , , ,

    We establish connections between the facial reduction algorithm of Borwein and Wolkowicz and the self-dual homogeneous model of Goldman and Tucker when applied to conic optimization problems. Specifically, we show the self-dual homogeneous model returns facial reduction certificates when it fails to return a primal-dual optimal solution or a certificate of infeasibility. Using this observation, … Read more

    Linear conic formulations for two-party correlations and values of nonlocal games

  • Jamie Sikora
  • Antonios Varvitsiotis
    • Categories Applications - Science and Engineering, Graphs and Matroids, Linear, Cone and Semidefinite Programming Tags , , , , ,

    In this work we study the sets of two-party correlations generated from a Bell scenario involving two spatially separated systems with respect to various physical models. We show that the sets of classical, quantum, no-signaling and unrestricted correlations can be expressed as projections of affine sections of appropriate convex cones. As a by-product, we identify … Read more

    Strong Duality: Without Simplex and without theorems of alternatives

  • Somdeb Lahiri
    • Categories Linear Programming Tags , ,

    The simplex method has its own problems related to degenerate basic feasible solutions. While such solutions are infrequent, from a theoretical standpoint a proof of the strong duality theorem that uses the simplex method is not complete until it has taken a few extra steps. Further, for economists the duality theorem is extremely important whereas … Read more

    A Stochastic Programming Approach for Shelter Location and Evacuation Planning

  • Vedat Bayram
  • Hande Yaman
    • Categories Second-Order Cone Programming, Stochastic Programming, Transportation Tags , , , , ,

    Shelter location and traffic allocation decisions are critical for an efficient evacuation plan. In this study, we propose a scenario-based two-stage stochastic evacuation planning model that optimally locates shelter sites and that assigns evacuees to nearest shelters and to shortest paths within a tolerance degree to minimize the expected total evacuation time. Our model considers … Read more

    Noisy Euclidean distance realization: robust facial reduction and the Pareto frontier

  • Dmitriy Drusvyatskiy
  • Nathan Krislock
  • Henry Wolkowicz
  • Y.-L. Voronin
    • Categories Convex Optimization, Semi-definite Programming Tags , , , , ,

    We present two algorithms for large-scale low-rank Euclidean distance matrix completion problems, based on semidefinite optimization. Our first method works by relating cliques in the graph of the known distances to faces of the positive semidefinite cone, yielding a combinatorial procedure that is provably robust and parallelizable. Our second algorithm is a first order method … Read more

    Inner Approximations of Completely Positive Reformulations of Mixed Binary Quadratic Programs: A Unified Analysis

  • E. Alper Yildirim
    • Categories Global Optimization, Linear, Cone and Semidefinite Programming Tags , , ,

    Every quadratic programming problem with a mix of continuous and binary variables can be equivalently reformulated as a completely positive optimization problem, i.e., a linear optimization problem over the convex but computationally intractable cone of completely positive matrices. In this paper, we focus on general inner approximations of the cone of completely positive matrices on … Read more

    The solution of Euclidean norm trust region SQP subproblems via second order cone programs, an overview and elementary introduction

  • Florian Jarre
  • Felix Lieder
    • Categories Constrained Nonlinear Optimization, Second-Order Cone Programming Tags , ,

    It is well known that convex SQP subproblems with a Euclidean norm trust region constraint can be reduced to second order cone programs for which the theory of Euclidean Jordan-algebras leads to efficient interior-point algorithms. Here, a brief and self-contained outline of the principles of such an implementation is given. All identities relevant for the … Read more

    Linear Rate Convergence of the Alternating Direction Method of Multipliers for Convex Composite Quadratic and Semi-Definite Programming

  • Deren Han
  • Defeng Sun
  • Liwei Zhang
    • Categories Convex and Nonsmooth Optimization, Linear, Cone and Semidefinite Programming, Nonlinear Optimization Tags , , , , ,

    In this paper, we aim to provide a comprehensive analysis on the linear rate convergence of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex composite optimization problems. Under a certain error bound condition, we establish the global linear rate of convergence for a more general semi-proximal ADMM with the dual steplength … Read more

    A priori bounds on the condition numbers in interior-point methods

  • Florian Jarre
    • Categories Linear Programming, Semi-definite Programming Tags , ,

    Interior-point methods are known to be sensitive to ill-conditioning and to scaling of the data. This paper presents new asymptotically sharp bounds on the condition numbers of the linear systems at each iteration of an interior-point method for solving linear or semidefinite programs and discusses a stopping test which leads to a problem-independent “a priori” … Read more

    Mixed Integer Second-Order Cone Programming for the Horizontal and Vertical Free-flight Planning Problem

  • Liana Amaya Moreno
  • Armin Fügenschuh
  • Anton Kaier
  • Amina Mollaysa
  • Swen Schlobach
  • Zhi Yuan
    • Categories Airline Optimization, Second-Order Cone Programming Tags , ,

    In the past, travel routes for civil passenger and cargo air traffic were aligned to the air traffic network (ATN). To resolve the network congestion problem, the free-flight system has recently been introduced in more and more regions around the globe, allowing flight operations to make full use of the four space-and-time dimensions. For the … Read more