Roberto Andreani

Convergence properties of a second order augmented Lagrangian method for mathematical programs with complementarity constraints

  • Roberto Andreani
  • Paulo J. S. Silva
  • Leonardo D. Secchin
    • Categories Complementarity and Variational Inequalities Tags , ,

    Mathematical Programs with Complementarity Constraints (MPCCs) are difficult optimization problems that do not satisfy the majority of the usual constraint qualifications (CQs) for standard nonlinear optimization. Despite this fact, classical methods behaves well when applied to MPCCs. Recently, Izmailov, Solodov and Uskov proved that first order augmented Lagrangian methods, under a natural adaption of the … Read more

    Strict Constraint Qualifications and Sequential Optimality Conditions for Constrained Optimization

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

    Sequential optimality conditions for constrained optimization are necessarily satisfied by local minimizers, independently of the fulfillment of constraint qualifications. These conditions support the employment of different stopping criteria for practical optimization algorithms. On the other hand, when an appropriate strict constraint qualification associated with some sequential optimality condition holds at a point that satisfies the … Read more

    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