Nonlinear Optimization

Iteration-complexity of an inexact proximal accelerated augmented Lagrangian method for solving linearly constrained smooth nonconvex composite optimization problems

  • Jefferson Gonçalves Melo
  • Renato D.C. Monteiro
  • Hairong Wang
    • Categories Constrained Nonlinear Optimization Tags , , ,

    This paper proposes and establishes the iteration-complexity of an inexact proximal accelerated augmented Lagrangian (IPAAL) method for solving linearly constrained smooth nonconvex composite optimization problems. Each IPAAL iteration consists of inexactly solving a proximal augmented Lagrangian subproblem by an accelerated composite gradient (ACG) method followed by a suitable Lagrange multiplier update. It is shown that … Read more

    Data Approximation by L1 Spline Fits with Free Knots

  • Ziteng Wang
  • Manfei Xie
    • Categories Applications - OR and Management Sciences, Applications - Science and Engineering, Nonlinear Optimization

    L1 spline fits are a class of spline models that have shown advantages in approximating irregular and multiscale data. This paper investigates the knot placement problem of L1 spline fits under two scenarios. If the number of knots is given, we propose an augmented Lagrangian method to solve the bilevel L1 spline fit problem and … Read more

    Optimality Conditions for Constrained Minimax Optimization

  • Yu-Hong Dai
  • Liwei Zhang
    • Categories Constrained Nonlinear Optimization Tags , , , , ,

    Minimax optimization problems arises from both modern machine learning including generative adversarial networks, adversarial training and multi-agent reinforcement learning, as well as from tradition research areas such as saddle point problems, numerical partial differential equations and optimality conditions of equality constrained optimization. For the unconstrained continuous nonconvex-nonconcave situation, Jin, Netrapalli and Jordan (2019) carefully considered … Read more

    DMulti-MADS: Mesh adaptive direct multisearch for blackbox multiobjective optimization

  • Jean Bigeon
  • Sébastien Le Digabel
  • Ludovic Salomon
    • Categories Multi-Criteria Optimization, Nonlinear Optimization Tags , , , ,

    The context of this research is multiobjective optimization where conflicting objectives are present. In this work, these objectives are only available as the outputs of a blackbox for which no derivative information is available. This work proposes a new extension of the mesh adaptive direct search (MADS) algorithm to constrained multiobjective derivative-free optimization. This method … Read more

    A primal-dual interior-point relaxation method with adaptively updating barrier for nonlinear programs

  • Yu-Hong Dai
  • Yakui Huang
  • Xin-Wei Liu
    • Categories Nonlinear Optimization Tags , , , ,

    Based on solving an equivalent parametric equality constrained mini-max problem of the classic logarithmic-barrier subproblem, we present a novel primal-dual interior-point relaxation method for nonlinear programs. In the proposed method, the barrier parameter is updated in every step as done in interior-point methods for linear programs, which is prominently different from the existing interior-point methods … Read more

    Approximate solution of system of equations arising in interior-point methods for bound-constrained optimization

  • David Ek
  • Anders Forsgren
    • Categories Bound-constrained Optimization Tags , , ,

    The focus in this paper is interior-point methods for bound-constrained nonlinear optimization where the system of nonlinear equations that arise are solved with Newton’s method. There is a trade-off between solving Newton systems directly, which give high quality solutions, and solving many approximate Newton systems which are computationally less expensive but give lower quality solutions. … Read more

    Stochastic Variance-Reduced Prox-Linear Algorithms for Nonconvex Composite Optimization

  • Lin Xiao
  • Junyu Zhang
    • Categories Nonlinear Optimization, Nonsmooth Optimization, Stochastic Programming Tags , , , ,

    We consider minimization of composite functions of the form $f(g(x))+h(x)$, where $f$ and $h$ are convex functions (which can be nonsmooth) and $g$ is a smooth vector mapping. In addition, we assume that $g$ is the average of finite number of component mappings or the expectation over a family of random component mappings. We propose … Read more

    Survey of Sequential Convex Programming and Generalized Gauss-Newton Methods

  • Katrin Baumgärtner
  • Moritz Diehl
  • Florian Messerer
    • Categories Constrained Nonlinear Optimization Tags , , , , ,

    We provide an overview of a class of iterative convex approximation methods for nonlinear optimization problems with convex-over-nonlinear substructure. These problems are characterized by outer convexities on the one hand, and nonlinear, generally nonconvex, but differentiable functions on the other hand. All methods from this class use only first order derivatives of the nonlinear functions … Read more

    Conditional gradient method for multiobjective optimization

  • Pedro Bonfim Assunção
  • Orizon Pereira Ferreira
  • Leandro Fonseca Prudente
    • Categories Constrained Nonlinear Optimization, Multi-Criteria Optimization

    We analyze the conditional gradient method, also known as Frank-Wolfe method, for constrained multiobjective optimization. The constraint set is assumed to be convex and compact, and the objectives functions are assumed to be continuously differentiable. The method is considered with different strategies for obtaining the step sizes. Asymptotic convergence properties and iteration-complexity bounds with and … Read more

    On monotonicity and search traversal in copositivity detection algorithms

  • Leocadio G Casado
  • Eligius M.T. Hendrix
  • Boglárka Tóth
    • Categories (Mixed) Integer Nonlinear Programming, Global Optimization, Quadratic Programming Tags , , ,

    Matrix copositivity has an important theoretical background. Over the last decades, the use of algorithms to check copositivity has made a big progress. Methods are based on spatial branch and bound, transformation to Mixed Integer Programming, implicit enumeration of KKT points or face-based search. Our research question focuses on exploiting the mathematical properties of the … Read more