Nonsmooth Optimization

An Interior-Point Method for Nonlinear Optimization Problems with Locatable and Separable Nonsmoothness

  • Martin Schmidt
    • Categories Applications - Science and Engineering, Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , , ,

    A lot of real-world optimization models comprise nonconvex and nonlinear as well as nonsmooth functions leading to very hard classes of optimization models. In this article a new interior-point method for the special but practically relevant class of optimization problems with locatable and separable nonsmooth aspects is presented. After motivating and formalizing the problems under … Read more

    An accelerated HPE-type algorithm for a class of composite convex-concave saddle-point problems

  • Yunlong He
  • Renato D.C. Monteiro
    • Categories Convex Optimization, Generalized Convexity/Monoticity, Nonsmooth Optimization Tags , , , , , , ,

    This article proposes a new algorithm for solving a class of composite convex-concave saddle-point problems. The new algorithm is a special instance of the hybrid proximal extragradient framework in which a Nesterov’s accelerated variant is used to approximately solve the prox subproblems. One of the advantages of the new method is that it works for … Read more

    Stochastic Quasi-Fejér Block-Coordinate Fixed Point Iterations with Random Sweeping

  • Patrick L. Combettes
  • Jean-Christophe Pesquet
    • Categories Convex Optimization, Nonsmooth Optimization, Stochastic Programming Tags , , , , , , ,

    This work investigates the properties of stochastic quasi-Fejér monotone sequences in Hilbert spaces and emphasizes their pertinence in the study of the convergence of block-coordinate fixed point methods. The iterative methods under investigation feature random sweeping rules to select the blocks of variables that are activated over the course of the iterations and allow for … Read more

    Forward – Backward Greedy Algorithms for Atomic – Norm Regularization

  • Nikhil Rao
  • Stephen Wright
  • Parikshit Shah
    • Categories Convex Optimization, Nonsmooth Optimization Tags , ,

    In many signal processing applications, one aims to reconstruct a signal that has a simple representation with respect to a certain basis or frame. Fundamental elements of the basis known as “atoms” allow us to define “atomic norms” that can be used to construct convex regularizers for the reconstruction problem. Efficient algorithms are available to … 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

    A Scalarization Proximal Point Method for Quasiconvex Multiobjective Minimization

  • H. C. F. Apolinário
  • Paulo Roberto Oliveira
  • Erik Alex Papa Quiroz
    • Categories Generalized Convexity/Monoticity, Multi-Criteria Optimization, Nonsmooth Optimization Tags , , , , ,

    In this paper we propose a scalarization proximal point method to solve multiobjective unconstrained minimization problems with locally Lipschitz and quasiconvex vector functions. We prove, under natural assumptions, that the sequence generated by the method is well defined and converges globally to a Pareto-Clarke critical point. Our method may be seen as an extension, for … Read more