Convex and Nonsmooth Optimization

Inexact Variable Metric Stochastic Block-Coordinate Descent for Regularized Optimization

  • Stephen Wright
  • Ching-pei Lee
    • Categories Convex Optimization, Nonsmooth Optimization

    Block-coordinate descent (BCD) is a popular framework for large-scale regularized optimization problems with block-separable structure. Existing methods have several limitations. They often assume that subproblems can be solved exactly at each iteration, which in practical terms usually restricts the quadratic term in the subproblem to be diagonal, thus losing most of the benefits of higher-order … Read more

    Generalized Stochastic Frank-Wolfe Algorithm with Stochastic “Substitute” Gradient for Structured Convex Optimization

  • Robert M. Freund
  • Haihao Lu
    • Categories Convex Optimization, Nonlinear Optimization, Stochastic Programming Tags , , ,

    The stochastic Frank-Wolfe method has recently attracted much general interest in the context of optimization for statistical and machine learning due to its ability to work with a more general feasible region. However, there has been a complexity gap in the guaranteed convergence rate for stochastic Frank-Wolfe compared to its deterministic counterpart. In this work, … Read more

    ACQUIRE: an inexact iteratively reweighted norm approach for TV-based Poisson image restoration

  • Daniela di Serafino
  • Germana Landi
  • Marco Viola
    • Categories Applications - Science and Engineering, Convex and Nonsmooth Optimization Tags , , , ,

    We propose a method, called ACQUIRE, for the solution of constrained optimization problems modeling the restoration of images corrupted by Poisson noise. The objective function is the sum of a generalized Kullback-Leibler divergence term and a TV regularizer, subject to nonnegativity and possibly other constraints, such as flux conservation. ACQUIRE is a line-search method that … Read more

    Minimizing convex quadratics with variable precision Krylov methods

  • Serge Gratton
  • Philippe L. Toint
  • Ehouarn Simon
    • Categories Convex Optimization, Optimization Software and Modeling Systems, Quadratic Programming Tags , ,

    Iterative algorithms for the solution of convex quadratic optimization problems are investigated, which exploit inaccurate matrix-vector products. Theoretical bounds on the performance of a Conjugate Gradients and a Full-Orthormalization methods are derived, the necessary quantities occurring in the theoretical bounds estimated and new practical algorithms derived. Numerical experiments suggest that the new methods have significant … Read more

    The Cyclic Douglas-Rachford Algorithm with r-sets-Douglas-Rachford Operators

  • Francisco J. Aragón Artacho
  • Yair Censor
  • Aviv Gibali
    • Categories Convex Optimization

    The Douglas-Rachford (DR) algorithm is an iterative procedure that uses sequential reflections onto convex sets and which has become popular for convex feasibility problems. In this paper we propose a structural generalization that allows to use r-sets-DR operators in a cyclic fashion. We prove convergence and present numerical illustrations of the potential advantage of such … Read more

    Characterizations of Differentiability, Smoothing Techniques and DC Programming with Applications to Image Reconstructions

  • Nguyen Thai An
  • Le Thi Hoai An
  • Daniel Giles
  • Nguyen Mau Nam
    • Categories Convex and Nonsmooth Optimization

    In this paper, we study characterizations of differentiability for real-valued functions based on generalized differentiation. These characterizations provide the mathematical foundation for Nesterov’s smoothing techniques in infinite dimensions. As an application, we provide a simple approach to image reconstructions based on Nesterov’s smoothing techniques and DC programming that involves the $\ell_1-\ell_2$ regularization. ArticleDownload View PDF

    Efficient Solution of Maximum-Entropy Sampling Problems

  • Kurt M. Anstreicher
    • Categories (Mixed) Integer Nonlinear Programming, Convex Optimization, Statistics Tags , ,

    We consider a new approach for the maximum-entropy sampling problem (MESP) that is based on bounds obtained by maximizing a function of the form ldet M(x) over linear constraints, where M(x)is linear in the n-vector x. These bounds can be computed very efficiently and are superior to all previously known bounds for MESP on most … Read more

    Finite convergence and weak sharpness for solutions of nonsmooth variational inequalities in Hilbert spaces

  • Qamrul Hasan Ansari
  • Luong Nguyen
  • Xiaolong Qin
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , , ,

    This paper deals with the study of weak sharp solutions for nonsmooth variational inequalities and finite convergence property of the proximal point method. We present several characterizations for weak sharpness of the solutions set of nonsmooth variational inequalities without using the gap functions. We show that under weak sharpness of the solutions set, the sequence … Read more

    A Unified Characterization of Nonlinear Scalarizing Functionals in Optimization

  • Gemayqzel Bouza
  • Christiane Tammer
  • Ernest Quintana
    • Categories Convex and Nonsmooth Optimization, Global Optimization, Nonlinear Optimization Tags , , , ,

    Over the years, several classes of scalarization techniques in optimization have been introduced and employed in deriving separation theorems, optimality conditions and algorithms. In this paper, we study the relationships between some of those classes in the sense of inclusion. We focus on three types of scalarizing functionals (by Hiriart-Urruty, Drummond and Svaiter, Gerstewitz) and … Read more

    The Standard Pessimistic Bilevel Problem

  • Lorenzo Lampariello
  • Simone Sagratella
  • Oliver Stein
    • Categories Constrained Nonlinear Optimization, Game Theory, Nonsmooth Optimization Tags , , ,

    Pessimistic bilevel optimization problems, as optimistic ones, possess a structure involving three interrelated optimization problems. Moreover, their finite infima are only attained under strong conditions. We address these difficulties within a framework of moderate assumptions and a perturbation approach which allow us to approximate such finite infima arbitrarily well by minimal values of a sequence … Read more