Constrained Nonlinear Optimization

Derivative-free methods for nonlinear programming with general lower-level constraints

  • Maria A. Diniz-Ehrhardt
  • José Mario Martínez
  • Lucas G. Pedroso
    • Categories Constrained Nonlinear Optimization Tags , , , , ,

    Augmented Lagrangian methods for derivative-free continuous optimization with constraints are introduced in this paper. The algorithms inherit the convergence results obtained by Andreani, Birgin, Martínez and Schuverdt for the case in which analytic derivatives exist and are available. In particular, feasible limit points satisfy KKT conditions under the Constant Positive Linear Dependence (CPLD) constraint qualification. … Read more

    A Penalty-Interior-Point Algorithm for Nonlinear Constrained Optimization

  • Frank E. Curtis
    • Categories Constrained Nonlinear Optimization Tags , , , , , ,

    Penalty and interior-point methods for nonlinear optimization problems have enjoyed great successes for decades. Penalty methods have proved to be effective for a variety of problem classes due to their regularization effects on the constraints. They have also been shown to allow for rapid infeasibility detection. Interior-point methods have become the workhorse in large-scale optimization … Read more

    Augmented Lagrangian method with nonmonotone penalty parameters for constrained optimization

  • Ernesto G. Birgin
  • José Mario Martínez
    • Categories Constrained Nonlinear Optimization Tags , , ,

    At each outer iteration of standard Augmented Lagrangian methods one tries to solve a box-constrained optimization problem with some prescribed tolerance. In the continuous world, using exact arithmetic, this subproblem is always solvable. Therefore, the possibility of finishing the subproblem resolution without satisfying the theoretical stopping conditions is not contemplated in usual convergence theories. However, … Read more

    A relaxed constant positive linear dependence constraint qualification and applications

  • Roberto Andreani
  • Gabriel Haeser
  • Paulo J. S. Silva
  • María Laura Schuverdt
    • Categories Constrained Nonlinear Optimization Tags , ,

    In this work we introduce a relaxed version of the constant positive linear dependence constraint qualification (CPLD) that we call RCPLD. This development is inspired by a recent generalization of the constant rank constraint qualification from Minchenko and Stakhovski that was called RCR. We show that RCPLD is enough to ensure the convergence of an … Read more

    Copositivity and constrained fractional quadratic problems

  • Paula Alexandra Amaral
  • Immanuel M. Bomze
  • Joaquim J. Júdice
    • Categories Constrained Nonlinear Optimization, Global Optimization Theory, Linear, Cone and Semidefinite Programming

    We provide Completely Positive and Copositive Programming formulations for the Constrained Fractional Quadratic Problem (CFQP) and Standard Fractional Quadratic Problem (StFQP). Based on these formulations, Semidefinite Programming (SDP) relaxations are derived for finding good lower bounds to these fractional programs, which are used in a global optimization branch-and-bound approach. Applications of the CFQP and StFQP, … Read more

    Local path-following property of inexact interior methods in nonlinear programming

  • Paul Armand
  • Jean-Pierre Dussault
  • Joël Benoist
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , , ,

    We study the local behavior of a primal-dual inexact interior point methods for solving nonlinear systems arising from the solution of nonlinear optimization problems or more generally from nonlinear complementarity problems. The algorithm is based on the Newton method applied to a sequence of perturbed systems that follows by perturbation of the complementarity equations of … Read more

    Perturbation resilience and superiorization of iterative algorithms

  • Yair Censor
  • Ran Davidi
  • Gabor T. Herman
    • Categories Biomedical Applications, Constrained Nonlinear Optimization Tags , , , ,

    Iterative algorithms aimed at solving some problems are discussed. For certain problems, such as finding a common point in the intersection of a finite number of convex sets, there often exist iterative algorithms that impose very little demand on computer resources. For other problems, such as finding that point in the intersection at which the … Read more

    On Equivalence of Semidefinite Relaxations for Quadratic Matrix Programming

  • Yichuan Ding
  • Dongdong Ge
  • Henry Wolkowicz
    • Categories Constrained Nonlinear Optimization, Convex Optimization Tags , , ,

    In this paper, we analyze two popular semidefinite programming \SDPb relaxations for quadratically constrained quadratic programs \QCQPb with matrix variables. These are based on \emph{vector-lifting} and on \emph{matrix lifting} and are of different size and expense. We prove, under mild assumptions, that these two relaxations provide equivalent bounds. Thus, our results provide a theoretical guideline … Read more

    Achieving Higher Frequencies in Large-Scale Nonlinear Model Predictive Control

  • Mihai Anitescu
  • Victor M. Zavala
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization, Control Applications

    We present new insights into how to achieve higher frequencies in large-scale nonlinear predictive control using truncated-like schemes. The basic idea is that, instead of solving the full nonlinear programming (NLP) problem at each sampling time, we solve a single, truncated quadratic programming (QP) problem. We present conditions guaranteeing stability of the approximation error derived … Read more

    A Gauss-Newton approach for solving constrained optimization problems using differentiable exact penalties

  • Roberto Andreani
  • Ellen H. Fukuda
  • Paulo J. S. Silva
    • Categories Constrained Nonlinear Optimization Tags , , ,

    We propose a Gauss-Newton-type method for nonlinear constrained optimization using the exact penalty introduced recently by Andre and Silva for variational inequalities. We extend their penalty function to both equality and inequality constraints using a weak regularity assumption, and as a result, we obtain a continuously differentiable exact penalty function and a new reformulation of … Read more