Linear, Cone and Semidefinite Programming

Long-Step Path-Following Algorithm for Solving Symmetric Programming Problems with Nonlinear Objective Functions

  • Leonid Faybusovich
  • Cunlu Zhou
    • Categories Convex Optimization, Linear, Cone and Semidefinite Programming Tags , ,

    We describe a long-step path-following algorithm for a class of symmetric programming problems with nonlinear convex objective functions. The complexity estimates similar to the case of a linear-quadratic objective function are established. The results of numerical experiments for the class of optimization problems involving quantum entropy are presented. Citation Preprint, University of Notre Dame, December … Read more

    On the local stability of semidefinite relaxations

  • Sameer Agarwal
  • Diego Cifuentes
  • Pablo A. Parrilo
  • Rekha R. Thomas
    • Categories Applications - Science and Engineering, Constrained Nonlinear Optimization, Semi-definite Programming Tags , , ,

    In this paper we consider a parametric family of polynomial optimization problems over algebraic sets. Although these problems are typically nonconvex, tractable convex relaxations via semidefinite programming (SDP) have been proposed. Often times in applications there is a natural value of the parameters for which the relaxation will solve the problem exactly. We study conditions … Read more

    Network-based Approximate Linear Programming for Discrete Optimization

  • Andre Augusto Cire
  • Selvaprabu Nadarajah
    • Categories (Mixed) Integer Nonlinear Programming, Dynamic Programming, Linear Programming Tags , ,

    We develop a new class of approximate linear programs (ALPs) that project the high-dimensional value function of dynamic programs onto a class of basis functions, each defined as a network that represents aggregrations over the state space. The resulting ALP is a minimum-cost flow problem over an extended variable space that synchronizes flows across multiple … Read more

    Amenable cones: error bounds without constraint qualifications

  • Bruno F. Lourenço
    • Categories Convex and Nonsmooth Optimization, Linear, Cone and Semidefinite Programming Tags , , , ,

    We provide a framework for obtaining error bounds for linear conic problems without assuming constraint qualifications or regularity conditions. The key aspects of our approach are the notions of amenable cones and facial residual functions. For amenable cones, it is shown that error bounds can be expressed as a composition of facial residual functions. The … Read more

    On the Convergence Rate of the Halpern-Iteration

  • Felix Lieder
    • Categories Linear, Cone and Semidefinite Programming, Nonlinear Optimization Tags , , ,

    In this work, we give a tight estimate of the rate of convergence for the Halpern-Iteration for approximating a fixed point of a nonexpansive mapping in a Hilbert space. Specifically, we prove that the norm of the residuals is upper bounded by the distance of the initial iterate to the closest fixed point divided by … Read more

    Set-Completely-Positive Representations and Cuts for the Max-Cut Polytope and the Unit Modulus Lifting

  • Florian Jarre
  • Ya-Feng Liu
  • Felix Lieder
  • Cheng Lu
    • Categories Convex Optimization, Linear, Cone and Semidefinite Programming Tags , , ,

    This paper considers a generalization of the “max-cut-polytope” $\conv\{\ xx^T\mid x\in\real^n, \ \ |x_k| = 1 \ \hbox{for} \ 1\le k\le n\}$ in the space of real symmetric $n\times n$-matrices with all-ones-diagonal to a complex “unit modulus lifting” $\conv\{xx\HH\mid x\in\complex^n, \ \ |x_k| = 1 \ \hbox{for} \ 1\le k\le n\}$ in the space of … Read more

    Weak Stability of $\ell_1hBcminimization Methods in Sparse Data Reconstruction

  • H. Jiang
  • Zhi-Quan Luo
  • Yun-Bin Zhao
    • Categories Convex and Nonsmooth Optimization, Data-Mining, Linear Programming Tags , , , , ,

    As one of the most plausible convex optimization methods for sparse data reconstruction, $\ell_1$-minimization plays a fundamental role in the development of sparse optimization theory. The stability of this method has been addressed in the literature under various assumptions such as restricted isometry property (RIP), null space property (NSP), and mutual coherence. In this paper, … Read more

    On Pathological Disjunctions and Redundant Disjunctive Conic Cuts

  • Julio C. Góez
  • Tamás Terlaky
  • Mohammad Shahabsafa
    • Categories Integer Programming, Second-Order Cone Programming Tags , ,

    The development of Disjunctive Conic Cuts (DCCs) for Mixed Integer Second Order Cone Optimization (MISOCO) problems has recently gained significant interest in the optimization community. In this paper, we explore the pathological disjunctions where disjunctive cuts do not tighten the description of the feasible set. We focus on the identification of cases when the generated … Read more

    Quadratic convergence to the optimal solution of second-order conic optimization without strict complementarity

  • Ali Mohammad-Nezhad
  • Tamás Terlaky
    • Categories Second-Order Cone Programming

    Under primal and dual nondegeneracy conditions, we establish the quadratic convergence of Newton’s method to the unique optimal solution of second-order conic optimization. Only very few approaches have been proposed to remedy the failure of strict complementarity, mostly based on nonsmooth analysis of the optimality conditions. Our local convergence result depends on the optimal partition … Read more

    Sieve-SDP: a simple facial reduction algorithm to preprocess semidefinite programs

  • Gabor Pataki
  • Quoc Tran-Dinh
  • Yuzixuan (Melody) Zhu
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Semi-definite Programming

    We introduce Sieve-SDP, a simple algorithm to preprocess semidefinite programs (SDPs). Sieve-SDP belongs to the class of facial reduction algorithms. It inspects the constraints of the problem, deletes redundant rows and columns, and reduces the size of the variable matrix. It often detects infeasibility. It does not rely on any optimization solver: the only subroutine … Read more