convex relaxation

Convex Quadratic Relaxations for Mixed-Integer Nonlinear Programs in Power Systems

  • Carleton Coffrin
  • Hassan L. Hijazi
  • Pascal Van Hentenryck
    • Categories Basic Sciences Applications, Convex Optimization, Global Optimization Applications Tags , , , , ,

    This paper presents a set of new convex quadratic relaxations for nonlinear and mixed-integer nonlinear programs arising in power systems. The considered models are motivated by hybrid discrete/continuous applications where existing approximations do not provide optimality guarantees. The new relaxations offer computational efficiency along with minimal optimality gaps, providing an interesting alternative to state-of-the-art semi-definite … Read more

    Convex relaxation for finding planted influential nodes in a social network

  • Lisa Elkin
  • Ting Kei Pong
  • Stephen A. Vavasis
    • Categories Convex Optimization, Data-Mining Tags , ,

    We consider the problem of maximizing influence in a social network. We focus on the case that the social network is a directed bipartite graph whose arcs join senders to receivers. We consider both the case of deterministic networks and probabilistic graphical models, that is, the so-called “cascade” model. The problem is to find the … Read more

    Robust convex relaxation for the planted clique and densest k-subgraph problems

  • Brendan Ames
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Nonsmooth Optimization Tags , , ,

    We consider the problem of identifying the densest k-node subgraph in a given graph. We write this problem as an instance of rank-constrained cardinality minimization and then relax using the nuclear and l1 norms. Although the original combinatorial problem is NP-hard, we show that the densest k-subgraph can be recovered from the solution of our … Read more

    Multi-Variate McCormick Relaxations

  • Alexander Mitsos
  • Angelos Tsoukalas
    • Categories Global Optimization Theory Tags , , , , , ,

    G. P. McCormick [Math Prog 1976] provides the framework for convex/concave relaxations of factorable functions, via rules for the product of functions and compositions of the form F(f(z)), where F is a univariate function. Herein, the composition theorem is generalized to allow multivariate outer functions F, and theory for the propagation of subgradients is presented. … Read more

    Old Wine in a New Bottle: The MILP Road to MIQCP

  • Samuel Burer
  • Anureet Saxena
    • Categories (Mixed) Integer Linear Programming, (Mixed) Integer Nonlinear Programming, Global Optimization Theory Tags , , ,

    This paper surveys results on the NP-hard mixed-integer quadratically constrained programming problem. The focus is strong convex relaxations and valid inequalities, which can become the basis of efficient global techniques. In particular, we discuss relaxations and inequalities arising from the algebraic description of the problem as well as from dynamic procedures based on disjunctive programming. … Read more

    Rank-Sparsity Incoherence for Matrix Decomposition

  • Venkat Chandrasekaran
  • Pablo A. Parrilo
  • Sujay Sanghavi
  • Alan S. Willsky
    • Categories Control Applications, Convex Optimization, Statistics Tags , , , , , , ,

    Suppose we are given a matrix that is formed by adding an unknown sparse matrix to an unknown low-rank matrix. Our goal is to decompose the given matrix into its sparse and low-rank components. Such a problem arises in a number of applications in model and system identification, and is NP-hard in general. In this … Read more

    On convex relaxations of quadrilinear terms

  • Sonia Cafieri
  • Jon Lee
  • Leo Liberti
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization Tags , , , , , , ,

    The best known method to find exact or at least epsilon-approximate solutions to polynomial programming problems is the spatial Branch-and-Bound algorithm, which rests on computing lower bounds to the value of the objective function to be minimized on each region that it explores. These lower bounds are often computed by solving convex relaxations of the … Read more

    Nuclear norm minimization for the planted clique and biclique problems

  • Brendan Ames
  • Stephen A. Vavasis
    • Categories Combinatorial Optimization, Convex Optimization Tags , , , , ,

    We consider the problems of finding a maximum clique in a graph and finding a maximum-edge biclique in a bipartite graph. Both problems are NP-hard. We write both problems as matrix-rank minimization and then relax them using the nuclear norm. This technique, which may be regarded as a generalization of compressive sensing, has recently been … Read more

    A Case Study of Joint Online Truck Scheduling and Inventory Management for Multiple Warehouses

  • Christoph Helmberg
  • Stefan Röhl
    • Categories (Mixed) Integer Linear Programming, Applications - OR and Management Sciences, Convex Optimization Tags , , , , , , ,

    For a real world problem — transporting pallets between warehouses in order to guarantee sufficient supply for known and additional stochastic demand — we propose a solution approach via convex relaxation of an integer programming formulation, suitable for online optimization. The essential new element linking routing and inventory management is a convex piecewise linear cost … Read more

    OPTIMIZATION-BASED SIMULATION OF NONSMOOTH RIGID MULTIBODY DYNAMICS

  • Mihai Anitescu
    • Categories Complementarity and Variational Inequalities, Mechanical Engineering Tags , , , ,

    We present a time-stepping method to simulate rigid multibody dynamics with inelastic collision, contact, and friction. The method progresses with fixed time step without backtracking for collision and solves at every step a strictly convex quadratic program. We prove that a solution sequence of the method converges to the solution of a measure differential inclusion. … Read more