optimality conditions

On the convergence of augmented Lagrangian strategies for nonlinear programming

  • Roberto Andreani
  • A. R. V. Cárdenas
  • Alberto Ramos
  • Ademir A. Ribeiro
  • Leonardo D. Secchin
    • Categories Nonlinear Optimization Tags , , , ,

    Augmented Lagrangian algorithms are very popular and successful methods for solving constrained optimization problems. Recently, the global convergence analysis of these methods have been dramatically improved by using the notion of the sequential optimality conditions. Such conditions are optimality conditions independently of the fulfilment of any constraint qualifications and provide theoretical tools to justify stopping … 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

    The Fermat Rule for Set Optimization Problems with Lipschitzian Set-Valued Mappings

  • Gemayqzel Bouza
  • Christiane Tammer
  • Ernest Quintana
  • Vu Anh Tuan
    • Categories Multi-Criteria Optimization, Nonsmooth Optimization, Robust Optimization Tags , , , ,

    n this paper, we consider set optimization problems with respect to the set approach. Specifically, we deal with the lower less and the upper less set relations. First, we derive properties of convexity and Lipschitzianity of suitable scalarizing functionals, under the same assumption on the set-valued objective mapping. We then obtain upper estimates of the … Read more

    Optimality conditions for nonlinear second-order cone programming and symmetric cone programming

  • Roberto Andreani
  • Ellen H. Fukuda
  • Gabriel Haeser
  • Daiana O. Santos
  • Leonardo D. Secchin
    • Categories Linear, Cone and Semidefinite Programming, Nonlinear Optimization, Second-Order Cone Programming Tags , , , ,

    Nonlinear symmetric cone programming (NSCP) generalizes important optimization problems such as nonlinear programming, nonlinear semidefinite programming and nonlinear second-order cone programming (NSOCP). In this work, we present two new optimality conditions for NSCP without constraint qualifications, which implies the Karush-Kuhn-Tucker conditions under a condition weaker than Robinson’s constraint qualification. In addition, we show the relationship … Read more

    Relations Between Abs-Normal NLPs and MPCCs Part 2: Weak Constraint Qualifications

  • Lisa Hegerhorst-Schultchen
  • Christian Kirches
  • Marc C. Steinbach
    • Categories Nonsmooth Optimization Tags , , , ,

    This work continues an ongoing effort to compare non-smooth optimization problems in abs-normal form to Mathematical Programs with Complementarity Constraints (MPCCs). We study general Nonlinear Programs with equality and inequality constraints in abs-normal form, so-called Abs-Normal NLPs, and their relation to equivalent MPCC reformulations. We introduce the concepts of Abadie’s and Guignard’s kink qualification and … Read more

    Relations Between Abs-Normal NLPs and MPCCs Part 1: Strong Constraint Qualifications

  • Lisa Hegerhorst-Schultchen
  • Christian Kirches
  • Marc C. Steinbach
    • Categories Nonsmooth Optimization Tags , , , ,

    This work is part of an ongoing effort of comparing non-smooth optimization problems in abs-normal form to MPCCs. We study the general abs-normal NLP with equality and inequality constraints in relation to an equivalent MPCC reformulation. We show that kink qualifications and MPCC constraint qualifications of linear independence type and Mangasarian-Fromovitz type are equivalent. Then … Read more

    Local minimizers of semi-algebraic functions

  • Tiên-So'n Pham
    • Categories Nonsmooth Optimization, Unconstrained Optimization Tags , , , , , ,

    Consider a semi-algebraic function $f\colon\mathbb{R}^n \to {\mathbb{R}},$ which is continuous around a point $\bar{x} \in \mathbb{R}^n.$ Using the so–called {\em tangency variety} of $f$ at $\bar{x},$ we first provide necessary and sufficient conditions for $\bar{x}$ to be a local minimizer of $f,$ and then in the case where $\bar{x}$ is an isolated local minimizer of … Read more

    Optimality conditions and global convergence for nonlinear semidefinite programming

  • Roberto Andreani
  • Gabriel Haeser
  • Daiana O. Santos
    • Categories Nonlinear Optimization, Semi-definite Programming Tags , , ,

    Sequential optimality conditions have played a major role in unifying and extending global convergence results for several classes of algorithms for general nonlinear optimization. In this paper, we extend theses concepts for nonlinear semidefinite programming. We define two sequential optimality conditions for nonlinear semidefinite programming. The first is a natural extension of the so-called Approximate-Karush-Kuhn-Tucker … Read more

    A Newton-CG Algorithm with Complexity Guarantees for Smooth Unconstrained Optimization

  • Michael J. O'Neill
  • Clément W. Royer
  • Stephen Wright
    • Categories Unconstrained Optimization Tags , , ,

    We consider minimization of a smooth nonconvex objective function using an iterative algorithm based on Newton’s method and linear conjugate gradient, with explicit detection and use of negative curvature directions for the Hessian of the objective function. The algorithm tracks Newton-conjugate gradient procedures developed in the 1980s closely, but includes enhancements that allow worst-case complexity … Read more