Nonlinear Optimization

On the Convergence and Complexity of Proximal Gradient and Accelerated Proximal Gradient Methods under Adaptive Gradient Estimation

  • Raghu Bollapragada
  • Shagun Gupta
    • Categories Nonlinear Optimization, Optimization in Data Science, Stochastic Programming Tags , , , ,

    In this paper, we propose a proximal gradient method and an accelerated proximal gradient method for solving composite optimization problems, where the objective function is the sum of a smooth and a convex, possibly nonsmooth, function. We consider settings where the smooth component is either a finite-sum function or an expectation of a stochastic function, … Read more

    Faster stochastic cubic regularized Newton methods with momentum

  • Yiming Yang
    • Categories Nonlinear Optimization, Optimization in Data Science, Unconstrained Optimization Tags , , ,

    Cubic regularized Newton (CRN) methods have attracted significant research interest because they offer stronger solution guarantees and lower iteration complexity. With the rise of the big-data era, there is growing interest in developing stochastic cubic regularized Newton (SCRN) methods that do not require exact gradient and Hessian evaluations. In this paper, we propose faster SCRN … Read more

    Sub-sampled Trust-Region Methods with Deterministic Worst-Case Complexity Guarantees

  • Max L. N. Gonçalves
  • Geovani Nunes Grapiglia
    • Categories Unconstrained Optimization Tags , , ,

    In this paper, we develop and analyze sub-sampled trust-region methods for solving finite-sum optimization problems. These methods employ subsampling strategies to approximate the gradient and Hessian of the objective function, significantly reducing the overall computational cost. We propose a novel adaptive procedure for deterministically adjusting the sample size used for gradient (or gradient and Hessian) … Read more

    Recursive Bound-Constrained AdaGrad with Applications to Multilevel and Domain Decomposition Minimization

  • Serge Gratton
  • Alena Kopanicakova
  • Philippe L. Toint
    • Categories Bound-constrained Optimization, Data Science Algorithms, Nonlinear Optimization Tags , , , , , ,

    Two OFFO (Objective-Function Free Optimization) noise tolerant algorithms are presented that handle bound constraints, inexact gradients and use second-order information when available. The first is a multi-level method exploiting a hierarchical description of the problem and the second is a domain-decomposition method covering the standard addditive Schwarz decompositions. Both are generalizations of the first-order AdaGrad … Read more

    A first-order method for nonconvex-nonconcave minimax problems under a local Kurdyka-Łojasiewicz condition

  • Zhaosong Lu
  • Xiangyuan Wang
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization, Other Topics Tags , , , ,

    We study a class of nonconvex–nonconcave minimax problems in which the inner maximization problem satisfies a local Kurdyka–Łojasiewicz (KL) condition that may vary with the outer minimization variable. In contrast to the global KL or Polyak–Łojasiewicz (PL) conditions commonly assumed in the literature—which are significantly stronger and often too restrictive in practice—this local KL condition … Read more

    General Perturbation Resilient Dynamic String-Averaging for Inconsistent Problems with Superiorization

  • Kay Barshad
  • Yair Censor
    • Categories Constrained Nonlinear Optimization, Convex Optimization Tags , , , , , , , , , , , , , , ,

    In this paper we introduce a General Dynamic String-Averaging (GDSA) iterative scheme and investigate its convergence properties in the inconsistent case, that is, when the input operators don’t have a common fixed point. The Dynamic String-Averaging Projection (DSAP) algorithm itself was introduced in an 2013 paper, where its strong convergence and bounded perturbation resilience were … Read more

    Complexity of normalized stochastic first-order methods with momentum under heavy-tailed noise

  • Chuan He
  • Zhaosong Lu
  • Defeng Sun
  • Zhanwang Deng
    • Categories Nonlinear Optimization, Optimization in Data Science, Unconstrained Optimization Tags , , ,

    In this paper, we propose practical normalized stochastic first-order methods with Polyak momentum, multi-extrapolated momentum, and recursive momentum for solving unconstrained optimization problems. These methods employ dynamically updated algorithmic parameters and do not require explicit knowledge of problem-dependent quantities such as the Lipschitz constant or noise bound. We establish first-order oracle complexity results for finding … Read more

    First-order methods for stochastic and finite-sum convex optimization with deterministic constraints

  • Zhaosong Lu
  • Yifeng Xiao
    • Categories Constrained Nonlinear Optimization, Convex Optimization, Stochastic Programming Tags , , , , , ,

    In this paper, we study a class of stochastic and finite-sum convex optimization problems with deterministic constraints. Existing methods typically aim to find an \(\epsilon\)-expectedly feasible stochastic optimal solution, in which the expected constraint violation and expected optimality gap are both within a prescribed tolerance ϵ. However, in many practical applications, constraints must be nearly … Read more

    Lipschitz Stability for a Class of Parametric Optimization Problems with Polyhedral Feasible Set Mapping

  • Diethard Klatte
    • Categories Complementarity and Variational Inequalities, Nonlinear Optimization, Nonsmooth Optimization Tags , , , , ,

    This paper is devoted to the Lipschitz analysis of the solution sets and optimal values for a class of parametric optimization problems involving a polyhedral feasible set mapping and a quadratic objective function with arametric linear part. Recall that a multifunction is said to be polyhedral if its graph is the union of finitely many polyhedral … Read more