Nonsmooth Optimization

New results on subgradient methods for strongly convex optimization problems with a unified analysis

  • Masaru Ito
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Nonsmooth Optimization Tags , , , , , ,

    We develop subgradient- and gradient-based methods for minimizing strongly convex functions under a notion which generalizes the standard Euclidean strong convexity. We propose a unifying framework for subgradient methods which yields two kinds of methods, namely, the Proximal Gradient Method (PGM) and the Conditional Gradient Method (CGM), unifying several existing methods. The unifying framework provides … Read more

    Convergence rates for forward-backward dynamical systems associated with strongly monotone inclusions

  • Radu Ioan Bot
  • Ernö Robert Csetnek
    • Categories Convex Optimization, Nonsmooth Optimization, Systems governed by Differential Equations Optimization Tags , , , ,

    We investigate the convergence rates of the trajectories generated by implicit first and second order dynamical systems associated to the determination of the zeros of the sum of a maximally monotone operator and a monotone and Lipschitz continuous one in a real Hilbert space. We show that these trajectories strongly converge with exponential rate to … Read more

    An optimal subgradient algorithm with subspace search for costly convex optimization problems

  • Masoud Ahookhosh
  • Arnold Neumaier
    • Categories Convex and Nonsmooth Optimization, Convex Optimization, Nonsmooth Optimization Tags , , , , , ,

    This paper presents an acceleration of the optimal subgradient algorithm OSGA \cite{NeuO} for solving convex optimization problems, where the objective function involves costly affine and cheap nonlinear terms. We combine OSGA with a multidimensional subspace search technique, which leads to low-dimensional problem that can be solved efficiently. Numerical results concerning some applications are reported. A … Read more

    A forward-backward-forward differential equation and its asymptotic properties

  • Sebastian Banert
  • Radu Ioan Bot
    • Categories Convex Optimization, Dynamic Programming, Nonsmooth Optimization Tags , , , ,

    In this paper, we approach the problem of finding the zeros of the sum of a maximally monotone operator and a monotone and Lipschitz continuous one in a real Hilbert space via an implicit forward-backward-forward dynamical system with nonconstant relaxation parameters and stepsizes of the resolvents. Besides proving existence and uniqueness of strong global solutions … Read more

    The cone condition and nonsmoothness in linear generalized Nash games

  • Oliver Stein
  • Nathan Sudermann-Merx
    • Categories Game Theory, Nonsmooth Optimization

    We consider linear generalized Nash games and introduce the so-called cone condition which characterizes the smoothness of the Nikaido-Isoda function under weak assumptions. The latter mapping arises from a reformulation of the generalized Nash equilibrium problem as a possibly nonsmooth optimization problem. Other regularity conditions like LICQ or SMFC(Q) are only sufficient for smoothness, but … Read more

    A weighted Mirror Descent algorithm for nonsmooth convex optimization problem

  • Duy V.N. Luong
  • Panos Parpas
  • Berc Rustem
  • Daniel Rueckert
    • Categories Convex Optimization, Nonsmooth Optimization Tags , , ,

    Large scale nonsmooth convex optimization is a common problem for a range of computational areas including machine learning and computer vision. Problems in these areas contain special domain structures and characteristics. Special treatment of such problem domains, exploiting their structures, can significantly improve the the computational burden. We present a weighted Mirror Descent method to … Read more

    A Nonmonotone Approach without Differentiability Test for Gradient Sampling Methods

  • Elias Salomão Helou
  • Sandra Augusta Santos
  • Lucas E. A. Simões
    • Categories Nonsmooth Optimization, Unconstrained Optimization Tags , , , ,

    Recently, optimization problems involving nonsmooth and locally Lipschitz functions have been subject of investigation, and an innovative method known as Gradient Sampling has gained attention. Although the method has shown good results for important real problems, some drawbacks still remain unexplored. This study suggests modifications to the gradient sampling class of methods in order to … Read more

    Second order forward-backward dynamical systems for monotone inclusion problems

  • Radu Ioan Bot
  • Ernö Robert Csetnek
    • Categories Convex Optimization, Nonsmooth Optimization, Systems governed by Differential Equations Optimization Tags , , , ,

    We begin by considering second order dynamical systems of the from $\ddot x(t) + \Gamma (\dot x(t)) + \lambda(t)B(x(t))=0$, where $\Gamma: {\cal H}\rightarrow{\cal H}$ is an elliptic bounded self-adjoint linear operator defined on a real Hilbert space ${\cal H}$, $B: {\cal H}\rightarrow{\cal H}$ is a cocoercive operator and $\lambda:[0,+\infty)\rightarrow [0,+\infty)$ is a relaxation function depending … Read more

    Activity Identification and Local Linear Convergence of Forward–Backward-type methods

  • Jalal Fadili
  • Jingwei Liang
  • Gabriel Peyré
    • Categories Convex and Nonsmooth Optimization, Nonsmooth Optimization Tags , , , ,

    In this paper, we consider a class of Forward–Backward (FB) splitting methods that includes several variants (e.g. inertial schemes, FISTA) for minimizing the sum of two proper convex and lower semi-continuous functions, one of which has a Lipschitz continuous gradient, and the other is partly smooth relatively to a smooth active manifold $\mathcal{M}$. We propose … Read more

    Inexact Proximal Point Methods for Quasiconvex Minimization on Hadamard Manifolds

  • Nancy Baygorrea
  • Nelson Maculan
  • Erik Alex Papa Quiroz
    • Categories Generalized Convexity/Monoticity, Global Optimization, Nonsmooth Optimization Tags , , , ,

    In this paper we present two inexact proximal point algorithms to solve minimization problems for quasiconvex objective functions on Hadamard manifolds. We prove that under natural assumptions the sequence generated by the algorithms are well defined and converge to critical points of the problem. We also present an application of the method to demand theory … Read more