proximal bundle method

Multi-cut stochastic approximation methods for solving stochastic convex composite optimization

  • Jiaming Liang
  • Honghao Zhang
  • Renato D.C. Monteiro
    • Categories Convex Optimization, Stochastic Programming Tags , , ,

    The development of a multi-cut stochastic approximation (SA) method for solving stochastic convex composite optimization (SCCO) problems has remained an open challenge. The difficulty arises from the fact that the stochastic multi-cut model, constructed as the pointwise maximum of individual stochastic linearizations, provides a biased estimate of the objective function, with the error being uncontrollable. … Read more

    On the Acceleration of Proximal Bundle Methods

  • David Fersztand
  • Xu Andy Sun
    • Categories Convex and Nonsmooth Optimization, Convex Optimization, Nonsmooth Optimization Tags , , ,

    The proximal bundle method (PBM) is a fundamental and computationally effective algorithm for solving nonsmooth optimization problems. In this paper, we present the first variant of the PBM for smooth objectives, achieving an accelerated convergence rate of \(\frac{1}{\sqrt{\epsilon}}\log(\frac{1}{\epsilon})\), where \(\epsilon\) is the desired accuracy. Our approach addresses an open question regarding the convergence guarantee of … Read more

    Unifying restart accelerated gradient and proximal bundle methods

  • Jiaming Liang
    • Categories Convex and Nonsmooth Optimization, Convex Optimization Tags , , , ,

    This paper presents a novel restarted version of Nesterov’s accelerated gradient method and establishes its optimal iteration-complexity for solving convex smooth composite optimization problems. The proposed restart accelerated gradient method is shown to be a specific instance of the accelerated inexact proximal point framework introduced in “An accelerated hybrid proximal extragradient method for convex optimization … Read more

    Primal-dual proximal bundle and conditional gradient methods for convex problems

  • Jiaming Liang
    • Categories Convex and Nonsmooth Optimization, Convex Optimization, Nonsmooth Optimization Tags , , , , ,

    This paper studies the primal-dual convergence and iteration-complexity of proximal bundle methods for solving nonsmooth problems with convex structures. More specifically, we develop a family of primal-dual proximal bundle methods for solving convex nonsmooth composite optimization problems and establish the iteration-complexity in terms of a primal-dual gap. We also propose a class of proximal bundle … Read more

    The Proximal Bundle Algorithm Under a Frank-Wolfe Perspective: an Improved Complexity Analysis

  • David Fersztand
  • Xu Andy Sun
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , ,

    The proximal bundle algorithm (PBA) is a fundamental and computationally effective algorithm for solving optimization problems with nonsmooth components. We investigate its convergence rate, focusing on composite settings where one function is smooth and the other is piecewise linear. We interpret a sequence of null steps of the PBA as a Frank-Wolfe algorithm on the … Read more

    Universal subgradient and proximal bundle methods for convex and strongly convex hybrid composite optimization

  • Vincent Guigues
  • Jiaming Liang
  • Renato D.C. Monteiro
    • Categories Convex Optimization, Nonsmooth Optimization Tags , , ,

    This paper develops two parameter-free methods for solving convex and strongly convex hybrid composite optimization problems, namely, a composite subgradient type method and a proximal bundle type method. Both functional and stationary complexity bounds for the two methods are established in terms of the unknown strong convexity parameter. To the best of our knowledge, the … Read more

    Proximal bundle methods for hybrid weakly convex composite optimization problems

  • Jiaming Liang
  • Renato D.C. Monteiro
  • Honghao Zhang
    • Categories Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

    This paper establishes the iteration-complexity of proximal bundle methods for solving hybrid (i.e., a blend of smooth and nonsmooth) weakly convex composite optimization (HWC-CO) problems. This is done in a unified manner by considering a proximal bundle framework (PBF) based on a generic bundle update framework which includes various well-known bundle update schemes. In contrast … Read more

    A single cut proximal bundle method for stochastic convex composite optimization

  • Jiaming Liang
  • Vincent Guigues
  • Renato D.C. Monteiro
    • Categories Convex and Nonsmooth Optimization, Stochastic Programming Tags , , ,

    This paper considers optimization problems where the objective is the sum of a function given by an expectation and a closed convex composite function, and proposes stochastic composite proximal bundle (SCPB) methods for solving it. Complexity guarantees are established for them without requiring knowledge of parameters associated with the problem instance. Moreover, it is shown … Read more

    A unified analysis of a class of proximal bundle methods for solving hybrid convex composite optimization problems

  • Jiaming Liang
  • Renato D.C. Monteiro
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , ,

    This paper presents a proximal bundle (PB) framework based on a generic bundle update scheme for solving the hybrid convex composite optimization (HCCO) problem and establishes a common iteration-complexity bound for any variant belonging to it. As a consequence, iteration-complexity bounds for three PB variants based on different bundle update schemes are obtained in the … Read more

    Optimal Convergence Rates for the Proximal Bundle Method

  • Mateo Díaz
  • Benjamin Grimmer
    • Categories Convex Optimization, Unconstrained Optimization Tags , , , ,

    We study convergence rates of the classic proximal bundle method for a variety of nonsmooth convex optimization problems. We show that, without any modification, this algorithm adapts to converge faster in the presence of smoothness or a Hölder growth condition. Our analysis reveals that with a constant stepsize, the bundle method is adaptive, yet it … Read more