Constrained Nonlinear Optimization

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

    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

    Constraint Qualifications for Karush-Kuhn-Tucker Conditions in Constrained Multiobjective Optimization

  • Gabriel Haeser
  • Alberto Ramos
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , ,

    The notion of a normal cone of a given set is paramount in optimization and variational analysis. In this work, we give a definition of a multiobjective normal cone which is suitable for studying optimality conditions and constraint qualifications for multiobjective optimization problems. A detailed study of the properties of the multiobjective normal cone is … Read more

    On optimality conditions for nonlinear conic programming

  • Roberto Andreani
  • Walter Gómez
  • Gabriel Haeser
  • Alberto Ramos
  • Leonardo M. Mito
    • Categories Constrained Nonlinear Optimization, Linear, Cone and Semidefinite Programming, Nonlinear Optimization Tags , , ,

    Sequential optimality conditions have played a major role in proving stronger global convergence results of numerical algorithms for nonlinear programming. Several extensions have been described in conic contexts, where many open questions have arisen. In this paper, we present new sequential optimality conditions in the context of a general nonlinear conic framework, which explains and … Read more

    Near-optimal analysis of univariate moment bounds for polynomial optimization

  • Monique Laurent
  • Lucas Slot
    • Categories Constrained Nonlinear Optimization, Semi-definite Programming

    We consider a recent hierarchy of upper approximations proposed by Lasserre (arXiv:1907.097784, 2019) for the minimization of a polynomial f over a compact set K⊆ℝn. This hierarchy relies on using the push-forward measure of the Lebesgue measure on K by the polynomial f and involves univariate sums of squares of polynomials with growing degrees 2r. … Read more

    An Outer-approximation Guided Optimization Approach for Constrained Neural Network Inverse Problems

  • Myun-Seok Cheon
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization

    This paper discusses an outer-approximation guided optimization method for constrained neural network inverse problems with rectified linear units. The constrained neural network inverse problems refer to an optimization problem to find the best set of input values of a given trained neural network in order to produce a predefined desired output in presence of constraints … Read more

    A Class of Smooth Exact Penalty Function Methods for Optimization Problems with Orthogonality Constraints

  • Nachuan Xiao
  • Xin Liu
  • Ya-xiang Yuan
    • Categories Basic Sciences Applications, Constrained Nonlinear Optimization Tags , ,

    Updating the augmented Lagrangian multiplier by closed-form expression yields efficient first-order infeasible approach for optimization problems with orthogonality constraints. Hence, parallelization becomes tractable in solving this type of problems. Inspired by this closed-form updating scheme, we propose an exact penalty function model with compact convex constraints (PenC). We show that PenC can act as an … Read more

    A bundle method for nonsmooth DC programming with application to chance-constrained problems

  • Welington de Oliveira
  • Sophie Demassey
  • Paul Javal
  • Hugo Morais
  • Bhargav Swaminathan
  • Wim van Ackooij
    • Categories Constrained Nonlinear Optimization, Nonsmooth Optimization Tags

    This work considers nonsmooth and nonconvex optimization problems whose objective and constraint functions are defined by difference-of-convex (DC) functions. We consider an infeasible bundle method based on the so-called improvement functions to compute critical points for problems of this class. Our algorithm neither employs penalization techniques nor solves subproblems with linearized constraints. The approach, which … Read more

    A primal-dual modified log-barrier method for inequality constrained nonlinear optimization

  • Joshua D. Griffin
  • Riadh Omheni
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , ,

    We present a primal-dual modified log-barrier algorithm to solve inequality constrained nonlinear optimization problems. Basically, the algorithm is a Newton-like method applied to a perturbation of the optimality system that follows from a reformulation of the initial problem by introducing a modified log-barrier function to handle inequality constraints. The algorithm uses an outer/inner iteration scheme … Read more