Convex Optimization

A One-Parameter Family of Middle Proximal ADMM for Constrained Separable Convex Optimization

  • Jianchao Bai
  • Hongchao Zhang
    • Categories Convex Optimization Tags , , , , , ,

    This work is devoted to studying a family of Middle Proximal Alternating Direction Method of Multipliers (MP-ADM) for solving multi-block constrained separable convex optimization. Such one-parameter family of MP-ADM combines both Jacobian and Gauss-Seidel types of alternating direction method, and proximal point techniques are only applied to the middle subproblems to promote the convergence. We … 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

    Adaptive Sampling Strategies for Stochastic Optimization

  • Raghu Bollapragada
  • Richard H. Byrd
  • Jorge Nocedal
    • Categories Convex Optimization, Nonlinear Optimization, Stochastic Programming Tags , ,

    In this paper, we propose a stochastic optimization method that adaptively controls the sample size used in the computation of gradient approximations. Unlike other variance reduction techniques that either require additional storage or the regular computation of full gradients, the proposed method reduces variance by increasing the sample size as needed. The decision to increase … Read more

    Convergence rates of accelerated proximal gradient algorithms under independent noise

  • Jiang Hao
  • Barrio Roberto
  • Cheng Lizhi
  • Sun Tao
    • Categories Convex Optimization, Nonlinear Optimization Tags , ,

    We consider an accelerated proximal gradient algorithm for the composite optimization with “independent errors” (errors little related with historical information) for solving linear inverse problems. We present a new inexact version of FISTA algorithm considering deterministic and stochastic noises. We prove some convergence rates of the algorithm and we connect it with the current existing … 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

    Exact worst-case convergence rates of the proximal gradient method for composite convex minimization

  • François Glineur
  • Julien M. Hendrickx
  • Adrien B. Taylor
    • Categories Convex Optimization, Nonsmooth Optimization

    We study the worst-case convergence rates of the proximal gradient method for minimizing the sum of a smooth strongly convex function and a non-smooth convex function whose proximal operator is available. We establish the exact worst-case convergence rates of the proximal gradient method in this setting for any step size and for different standard performance … Read more

    Complete Facial Reduction in One Step for Spectrahedra

  • Stefan Sremac
  • Henry Wolkowicz
  • Hugo J. Woerdeman
    • Categories Convex Optimization, Semi-definite Programming Tags , , ,

    A spectrahedron is the feasible set of a semidefinite program, SDP, i.e., the intersection of an affine set with the positive semidefinite cone. While strict feasibility is a generic property for random problems, there are many classes of problems where strict feasibility fails and this means that strong duality can fail as well. If the … Read more

    Generalized ADMM with Optimal Inde nite Proximal Term for Linearly Constrained Convex Optimization

  • Xingju Cai
  • Fan Jiang
  • Zhongming Wu
    • Categories Convex Optimization Tags , , , , ,

    We consider the generalized alternating direction method of multipliers (ADMM) for linearly constrained convex optimization. Many problems derived from practical applications have showed that usually one of the subproblems in the generalized ADMM is hard to solve, thus a special proximal term is added. In the literature, the proximal term can be inde nite which plays … Read more

    Relative-Continuity” for Non-Lipschitz Non-Smooth Convex Optimization using Stochastic (or Deterministic) Mirror Descent

  • Haihao Lu
    • Categories Convex Optimization, Nonsmooth Optimization Tags , , ,

    The usual approach to developing and analyzing first-order methods for non-smooth (stochastic or deterministic) convex optimization assumes that the objective function is uniformly Lipschitz continuous with parameter $M_f$. However, in many settings the non-differentiable convex function $f(\cdot)$ is not uniformly Lipschitz continuous — for example (i) the classical support vector machine (SVM) problem, (ii) the … Read more

    Response to “Counterexample to global convergence of DSOS and SDSOS hierarchies”

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

    In a recent note [8], the author provides a counterexample to the global convergence of what his work refers to as “the DSOS and SDSOS hierarchies” for polynomial optimization problems (POPs) and purports that this refutes claims in our extended abstract [4] and slides in [3]. The goal of this paper is to clarify that … Read more