Linear, Cone and Semidefinite Programming

The Role of Level-Set Geometry on the Performance of PDHG for Conic Linear Optimization

  • Zikai Xiong
  • Robert M. Freund
    • Categories Convex Optimization, Linear Programming, Linear, Cone and Semidefinite Programming Tags , , , , ,

    We consider solving huge-scale instances of (convex) conic linear optimization problems, at the scale where matrix-factorization-free methods are attractive or necessary. The restarted primal-dual hybrid gradient method (rPDHG) — with heuristic enhancements and GPU implementation — has been very successful in solving huge-scale linear programming (LP) problems; however its application to more general conic convex … Read more

    Tighter yet more tractable relaxations and nontrivial instance generation for sparse standard quadratic optimization

  • Immanuel M. Bomze
  • Bo Peng
  • Yuzhou Qiu
  • E. Alper Yildirim
    • Categories (Mixed) Integer Nonlinear Programming, Cone Programming, Semi-definite Programming Tags , , , ,

    The Standard Quadratic optimization Problem (StQP), arguably the simplest among all classes of NP-hard optimization problems, consists of extremizing a quadratic form (the simplest nonlinear polynomial) over the standard simplex (the simplest polytope/compact feasible set). As a problem class, StQPs may be nonconvex with an exponential number of inefficient local solutions. StQPs arise in a … Read more

    Exploiting cone approximations in an augmented Lagrangian method for conic optimization

  • Mituhiro Fukuda
  • Walter Gómez
  • Gabriel Haeser
  • Leonardo M. Mito
    • Categories Cone Programming, Constrained Nonlinear Optimization Tags , , ,

    We propose an algorithm for general nonlinear conic programming which does not require the knowledge of the full cone, but rather a simpler, more tractable, approximation of it. We prove that the algorithm satisfies a strong global convergence property in the sense that it generates a strong sequential optimality condition. In particular, a KKT point … Read more

    On the integrality gap of the Complete Metric Steiner Tree Problem via a novel formulation

  • Ambrogio Maria Bernardelli
  • Eleonora Vercesi
  • Stefano Gualandi
  • Monaldo Mastrolilli
  • Luca Maria Gambardella
    • Categories Combinatorial Optimization, Graphs and Matroids, Linear Programming Tags , ,

    In this work, we study the metric Steiner Tree problem on graphs focusing on computing lower bounds for the integrality gap of the bi-directed cut (DCUT) formulation and introducing a novel formulation, the Complete Metric (CM) model, specifically designed to address the weakness of the DCUT formulation on metric instances. A key contribution of our … Read more

    Counterfactual Explanations for Linear Optimization

  • Jannis Kurtz
  • Ş. İlker Birbil
  • Dick den Hertog
    • Categories Applications - OR and Management Sciences, Linear Programming Tags , , ,

    The concept of counterfactual explanations (CE) has emerged as one of the important concepts to understand the inner workings of complex AI systems. In this paper, we translate the idea of CEs to linear optimization and propose, motivate, and analyze three different types of CEs: strong, weak, and relative. While deriving strong and weak CEs … Read more

    Sensitivity Analysis in Dantzig-Wolfe Decomposition

  • Kouhei Harada
    • Categories Linear Programming Tags , , ,

    Dantzig-Wolfe decomposition is a well-known classical method for solving huge linear optimization problems with a block-angular structure. The most computationally expensive process in the method is pricing: solving block subproblems for a dual variable to produce new columns. Therefore, when we want to solve a slightly perturbated problem in which the block-angular structure is preserved … Read more

    Exploiting Sign Symmetries in Minimizing Sums of Rational Functions

  • Jie Wang
  • Feng Guo
    • Categories Constrained Nonlinear Optimization, Global Optimization Applications, Semi-definite Programming Tags , , , ,

    This paper is devoted to the problem of minimizing a sum of rational functions over a basic semialgebraic set. We provide a hierarchy of sum of squares (SOS) relaxations that is dual to the generalized moment problem approach due to Bugarin, Henrion, and Lasserre. The investigation of the dual SOS aspect offers two benefits: 1) … Read more

    Edge expansion of a graph: SDP-based computational strategies

  • Akshay Gupte
  • Melanie Siebenhofer
  • Angelika Wiegele
    • Categories Combinatorial Optimization, Parallel Algorithms, Semi-definite Programming Tags , , , ,

    Computing the edge expansion of a graph is a famously hard combinatorial problem for which there have been many approximation studies. We present two variants of exact algorithms using semidefinite programming (SDP) to compute this constant for any graph. The first variant uses the SDP relax- ation first to reduce the search space considerably. One … Read more

    A Clustering-based uncertainty set for Robust Optimization

  • Alireza Yazdani
  • Ahmadreza Marandi
  • Rob Basten
  • Lijia Tan
    • Categories (Mixed) Integer Nonlinear Programming, Cone Programming, Robust Optimization Tags , , , ,

    Robust Optimization (RO) is an approach to tackle uncertainties in the parameters of an optimization problem. Constructing an uncertainty set is crucial for RO, as it determines the quality and the conservativeness of the solutions. In this paper, we introduce an approach for constructing a data-driven uncertainty set through volume-based clustering, which we call Minimum-Volume … Read more