Roberto Andreani

A second-order sequential optimality condition associated to the convergence of optimization algorithms

  • Roberto Andreani
  • Gabriel Haeser
  • Alberto Ramos
  • Paulo J. S. Silva
    • Categories Constrained Nonlinear Optimization Tags , ,

    Sequential optimality conditions have recently played an important role on the analysis of the global convergence of optimization algorithms towards first-order stationary points and justifying their stopping criteria. In this paper we introduce the first sequential optimality condition that takes into account second-order information. We also present a companion constraint qualification that is less stringent … Read more

    A cone-continuity constraint qualification and algorithmic consequences

  • Roberto Andreani
  • José Mario Martínez
  • Alberto Ramos
  • Paulo J. S. Silva
    • Categories Constrained Nonlinear Optimization Tags , , , ,

    Every local minimizer of a smooth constrained optimization problem satisfies the sequential Approximate Karush-Kuhn-Tucker (AKKT) condition. This optimality condition is used to define the stopping criteria of many practical nonlinear programming algorithms. It is natural to ask for conditions on the constraints under which AKKT implies KKT. These conditions will be called Strict Constraint Qualifications … Read more

    On Second Order Optimality Conditions in Nonlinear Optimization

  • Roberto Andreani
  • Roger Behling
  • Gabriel Haeser
  • Paulo J. S. Silva
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization

    In this work we present new weak conditions that ensure the validity of necessary second order optimality conditions (SOC) for nonlinear optimization. We are able to prove that weak and strong SOCs hold for all Lagrange multipliers using Abadie-type assumptions. We also prove weak and strong SOCs for at least one Lagrange multiplier imposing the … Read more

    Constant rank constraint qualifications: a geometric introduction

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

    Constraint qualifications (CQ) are assumptions on the algebraic description of the feasible set of an optimization problem that ensure that the KKT conditions hold at any local minimum. In this work we show that constraint qualifications based on the notion of constant rank can be understood as assumptions that ensure that the polar of the … Read more

    Two new weak constraint qualifications and applications

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

    We present two new constraint qualifications (CQ) that are weaker than the recently introduced Relaxed Constant Positive Linear Depen- dence (RCPLD) constraint qualification. RCPLD is based on the assump- tion that many subsets of the gradients of the active constraints preserve positive linear dependence locally. A major open question was to identify the exact set … 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

    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

    A new sequential optimality condition for constrained optimization and algorithmic consequences

  • Roberto Andreani
  • José Mario Martínez
  • Benar Fux Svaiter
    • Categories Nonlinear Optimization Tags , , ,

    Necessary first-order sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. These conditions are satisfied by local minimizers of optimization problems independently of the fulfillment of constraint qual- i cations. A new strong sequential optimality condition is introduced in the present paper. A proof that a well established Augmented Lagrangian … Read more

    On sequential optimality conditions for smooth constrained optimization

  • Roberto Andreani
  • Gabriel Haeser
  • José Mario Martínez
    • Categories Nonlinear Optimization Tags , , ,

    Sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. Approximate KKT and Approximate Gradient Projection conditions are analyzed in this work. These conditions are not necessarily equivalent. Implications between different conditions and counter-examples will be shown. Algorithmic consequences will be discussed. Article Download View On sequential optimality conditions for … Read more

    The Constant Rank Condition and Second Order Constraint Qualifications

  • Roberto Andreani
  • Carlos Eugenio Echagüe
  • María Laura Schuverdt
    • Categories Nonlinear Optimization Tags ,

    The Constant Rank condition for feasible points of nonlinear programming problems was defined by Janin in Ref. 1. In that paper the author proved that the condition was a first order constraint qualification. In this work we prove that the Janin Constant Rank condition is, in addition, a second order constraint qualification. We also define … Read more