Convex and Nonsmooth Optimization

A note on Fejér-monotone sequences in product spaces and its applications to the dual convergence of augmented Lagrangian methods

  • M. Marques Alves
  • Benar Fux Svaiter
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , ,

    In a recent Math. Program. paper, Eckstein and Silva proposed a new error criterion for the approximate solutions of augmented Lagrangian subproblems. Based on a saddle-point formulation of the primal and dual problems, they proved that dual sequences generated by augmented Lagrangians under this error criterion are bounded and that theirs limit points are dual … Read more

    A Generalized Inexact Proximal Point Method for Nonsmooth Functions that Satisfies Kurdyka Lojasiewicz Inequality

  • Glaydston Bento
  • Antoine Soubeyran
    • Categories Nonsmooth Optimization

    In this paper, following the ideas presented in Attouch et al. (Math. Program. Ser. A, 137: 91-129, 2013), we present an inexact version of the proximal point method for nonsmoth functions, whose regularization is given by a generalized perturbation term. More precisely, the new perturbation term is defined as a “curved enough” function of the … Read more

    Parallel Algorithms for Big Data Optimization

  • Francisco Facchinei
  • Simone Sagratella
  • Gesualdo Scutari
    • Categories Nonlinear Optimization, Nonsmooth Optimization, Parallel Algorithms Tags , , , ,

    We propose a decomposition framework for the parallel optimization of the sum of a differentiable function and a (block) separable nonsmooth, convex one. The latter term is usually employed to enforce structure in the solution, typically sparsity. Our framework is very flexible and includes both fully parallel Jacobi schemes and Gauss-Seidel (i.e., sequential) ones, as … Read more

    Dynamic scaling in the Mesh Adaptive Direct Search algorithm for blackbox optimization

  • Charles Audet
  • Sébastien Le Digabel
  • Christophe Tribes
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , ,

    Blackbox optimization deals with situations in which the objective function and constraints are typically computed by launching a time-consuming computer sim- ulation. The subject of this work is the Mesh Adaptive Direct Search (MADS) class of algorithms for blackbox optimization. We propose a way to dynamically scale the mesh, which is the discrete spatial structure … Read more

    Generalized Inexact Proximal Algorithms: Habit’s/ Routine’s Formation with Resistance to Change, following Worthwhile Changes

  • Glaydston Bento
  • Antoine Soubeyran
    • Categories Applications - Science and Engineering, Nonsmooth Optimization, Unconstrained Optimization Tags

    This paper shows how, in a quasi metric space, an inexact proximal algorithm with a generalized perturbation term appears to be a nice tool for Behavioral Sciences (Psychology, Economics, Management, Game theory,…). More precisely, the new perturbation term represents an index of resistance to change, defined as a “curved enough” function of the quasi distance … Read more

    Fixed points and variational principles with applications to capability theory of wellbeing via variational rationality

  • Boris S. Mordukhovich
  • Antoine Soubeyran
  • Bao Q. Truong
    • Categories Applications - OR and Management Sciences, Multi-Criteria Optimization, Nonsmooth Optimization Tags , , , ,

    In this paper we first develop two new results of variational analysis. One is a fixed point theorem for parametric dynamic systems in quasimetric spaces, which can also be interpreted as an existence theorem of minimal points with respect to reflexive and transitive preferences for sets in products spaces. The other one is a variational … Read more

    On the Direct Extension of ADMM for Multi-block Separable Convex Programming and Beyond: From Variational Inequality Perspective

  • Bingsheng He
  • Xiao-Ming Yuan
    • Categories Convex Optimization Tags , , , , ,

    When the alternating direction method of multipliers (ADMM) is extended directly to a multi-block separable convex minimization model whose objective function is in form of more than two functions without coupled variables, it was recently shown that the convergence is not guaranteed. This fact urges to develop efficient algorithms that can preserve completely the numerical … Read more

    Finding the largest low-rank clusters with Ky Fan 2-k-norm and l1-norm

  • Xuan Vinh Doan
  • Stephen A. Vavasis
    • Categories Convex Optimization, Data-Mining Tags , ,

    We propose a convex optimization formulation with the Ky Fan 2-k-norm and l1-norm to fi nd k largest approximately rank-one submatrix blocks of a given nonnegative matrix that has low-rank block diagonal structure with noise. We analyze low-rank and sparsity structures of the optimal solutions using properties of these two matrix norms. We show that, under … Read more

    About the Convexity of a Special Function on Hadamard Manifolds.

  • João Xavier da Cruz Neto
  • I.D. Melo
  • J.P. Silva
  • P.A. Sousa
    • Categories Generalized Convexity/Monoticity

    In this article we provide an erratum to Proposition 3.4 of E.A. Papa Quiroz and P.R. Oliveira. Proximal Point Methods for Quasiconvex and Convex Functions with Bregman Distances on Hadamard Manifolds, Journal of Convex Analysis 16 (2009), 49-69. More specifically, we prove that the function defined by the product of a fixed vector in the … Read more

    Direct search based on probabilistic descent

  • Serge Gratton
  • Clément W. Royer
  • Luis Nunes Vicente
  • Z. Zhang
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization, Stochastic Programming Tags , , , , ,

    Direct-search methods are a class of popular derivative-free algorithms characterized by evaluating the objective function using a step size and a number of (polling) directions. When applied to the minimization of smooth functions, the polling directions are typically taken from positive spanning sets which in turn must have at least n+1 vectors in an n-dimensional … Read more