Nonsmooth Optimization

Approaches to iterative algorithms for solving nonlinear equations with an application in tomographic absorption spectroscopy

  • Francisco J. Aragón Artacho
  • Weiwei Cai
  • Yair Censor
  • Aviv Gibali
  • Chongyuan Shui
  • David Torregrosa-Belén
    • Categories Biomedical Applications, Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , , , ,

    In this paper we propose an approach for solving systems of nonlinear equations without computing function derivatives. Motivated by the application area of tomographic absorption spectroscopy, which is a highly-nonlinear problem with variables coupling, we consider a situation where straightforward translation to a fixed point problem is not possible because the operators that represent the … Read more

    Understanding the Douglas-Rachford splitting method through the lenses of Moreau-type envelopes

  • Felipe Atenas
    • Categories Convex and Nonsmooth Optimization, Generalized Convexity/Monoticity, Nonsmooth Optimization Tags , , , ,

    We analyze the Douglas-Rachford splitting method for weakly convex optimization problems, by the token of the Douglas-Rachford envelope, a merit function akin to the Moreau envelope. First, we use epi-convergence techniques to show that this artifact approximates the original objective function via epigraphs. Secondly, we present how global convergence and local linear convergence rates for … Read more

    Second-Order Strong Optimality and Second-Order Duality for Nonsmooth Constrained Multiobjective Fractional Programming Problems

  • Chen Jiawei
  • Luyu Liu
  • Jen-Chih Yao
    • Categories Multi-Criteria Optimization, Nonsmooth Optimization Tags , , , ,

    This paper investigates constrained nonsmooth multiobjective fractional programming problem (NMFP) in real Banach spaces. It derives a quotient calculus rule for computing the first- and second-order Clarke derivatives of fractional functions involving locally Lipschitz functions. A novel second-order Abadie-type regularity condition is presented, defined with the help of the Clarke directional derivative and the P´ales-Zeidan … Read more

    Fast convergence of the primal-dual dynamical system and algorithms for a nonsmooth bilinearly coupled saddle point problem

  • Kewei Ding
  • Jörg Fliege
  • Phan Vuong
    • Categories Game Theory, Nonsmooth Optimization Tags , , , ,

    This paper is devoted to study the convergence rates of a second-order dynamical system and its corresponding discretizations associated with a nonsmooth bilinearly coupled convex-concave saddle point problem. We derive the convergence rate of the primal-dual gap for the second-order dynamical system with asymptotically vanishing damping term. Based on the implicit discretization, we propose a … Read more

    Scalable Projection-Free Optimization Methods via MultiRadial Duality Theory

  • Thabo Samakhoana
  • Benjamin Grimmer
    • Categories Convex and Nonsmooth Optimization, Convex Optimization, Nonsmooth Optimization Tags , , ,

    Recent works have developed new projection-free first-order methods based on utilizing linesearches and normal vector computations to maintain feasibility. These oracles can be cheaper than orthogonal projection or linear optimization subroutines but have the drawback of requiring a known strictly feasible point to do these linesearches with respect to. In this work, we develop new … Read more

    Some Primal-Dual Theory for Subgradient Methods for Strongly Convex Optimization

  • Benjamin Grimmer
  • Danlin Li
    • Categories Convex and Nonsmooth Optimization, Convex Optimization, Nonsmooth Optimization Tags , , , , ,

    We consider (stochastic) subgradient methods for strongly convex but potentially nonsmooth non-Lipschitz optimization. We provide new equivalent dual descriptions (in the style of dual averaging) for the classic subgradient method, the proximal subgradient method, and the switching subgradient method. These equivalences enable $O(1/T)$ convergence guarantees in terms of both their classic primal gap and a … Read more

    Shadow splitting methods for nonconvex optimisation: epi-approximation, convergence and saddle point avoidance

  • Felipe Atenas
    • Categories Generalized Convexity/Monoticity, Nonsmooth Optimization Tags , , , , , ,

    We propose the shadow Davis-Yin three-operator splitting method to solve nonconvex optimisation problems. Its convergence analysis is based on a merit function resembling the Moreau envelope. We explore variational analysis properties behind the merit function and the iteration operators associated with the shadow method. By capitalising on these results, we establish convergence of a damped … Read more

    Weak convexity and approximate subdifferentials

  • Felipe Atenas
  • Wim van Ackooij
  • Claudia Sagastizábal
    • Categories Complementarity and Variational Inequalities, Generalized Convexity/Monoticity, Nonsmooth Optimization

    We explore and construct an enlarged subdifferential for weakly convex functions. The resulting object turns out to be continuous with respect to both the function argument and the enlargement parameter. We carefully analyze connections with other constructs in the literature and particularize to the weakly convex setting well-known variational principles. By resorting to the new … Read more

    An Inexact Restoration Direct Multisearch Filter Approach to Multiobjective Constrained Derivative-free Optimization

  • Everton Silva
  • Ana Luisa Custódio
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , , , ,

    Direct Multisearch (DMS) is a well-established class of methods for multiobjective derivative-free optimization, where constraints are addressed by an extreme barrier approach, only evaluating feasible points. In this work, we propose a filter approach, combined with an inexact feasibility restoration step, to address constraints in the DMS framework. The filter approach treats feasibility as an … Read more

    Solving separable convex optimization problems: Faster prediction-correction framework

  • Tao Zhang
    • Categories Convex and Nonsmooth Optimization, Convex Optimization, Nonsmooth Optimization

    He and Yuan’s prediction-correction framework [SIAM J. Numer. Anal. 50: 700-709, 2012] is able to provide convergent algorithms for solving separable convex optimization problems at a rate of $O(1/t)$ ($t$ represents iteration times) in both ergodic (the average of iteration) and pointwise senses. This paper presents a faster prediction-correction framework at a rate of $O(1/t)$ … Read more