abs-normal form

How Stringent is the Linear Independence Kink Qualification in Abs-Smooth Optimization?

  • Franz Bethke
  • Andrea Walther
    • Categories Nonsmooth Optimization Tags

    Abs-smooth functions are given by compositions of smooth functions and the evaluation of the absolute value. The linear independence kink qualification (LIKQ) is a fundamental assumption in optimization problems governed by these abs-smooth functions, generalizing the well-known LICQ from smooth optimization. In particular, provided that LIKQ holds it is possible to derive optimality conditions for … 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

    On First and Second Order Optimality Conditions for Abs-Normal NLP

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

    Structured nonsmoothness is widely present in practical optimization. A particularly attractive class of nonsmooth problems, both from a theoretical and from an algorithmic perspective, are optimization problems in so-called abs-normal form as developed by Griewank and Walther. Here we generalize their theory for the unconstrained case to nonsmooth NLPs with equality and inequality constraints in … Read more

    On the Relation between MPECs and Optimization Problems in Abs-Normal Form

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

    We show that the problem of unconstrained minimization of a function in abs-normal form is equivalent to identifying a certain stationary point of a counterpart Mathematical Program with Equilibrium Constraints (MPEC). Hence, concepts introduced for the abs-normal forms turn out to be closely related to established concepts in the theory of MPECs. We give a … Read more

    An Algorithm for Piecewise Linear Optimization of Objective Functions in Abs-normal Form

  • Andreas Griewank
  • Andrea Walther
    • Categories Nonsmooth Optimization Tags , , , , , , ,

    In the paper [11] we derived first order (KKT) and second order (SSC) optimality conditions for functions defined by evaluation programs involving smooth elementals and absolute values. For this class of problems we showed in [12] that the natural algorithm of successive piecewise linear optimization with a proximal term (SPLOP) achieves a linear or even … Read more

    Relaxing kink qualifications and proving convergence rates in piecewise smooth optimization

  • Andreas Griewank
  • Andrea Walther
    • Categories Nonsmooth Optimization Tags , , , , , , ,

    Abstract. In the paper [9] we derived first order (KKT) and second order (SSC) optimality conditions for functions defined by evaluation programs involving smooth elementals and absolute values. In that analysis, a key assumption on the local piecewise linearization was the Linear Independence Kink Qualification (LIKQ), a generalization of the Linear Independence Constraint Qualification (LICQ) … Read more

    Characterizing and testing subdifferential regularity for piecewise smooth objective functions

  • Andreas Griewank
  • Andrea Walther
    • Categories Nonsmooth Optimization Tags , , , , , ,

    Functions defined by evaluation programs involving smooth elementals and absolute values as well as the max- and min-operator are piecewise smooth. Using piecewise linearization we derived in [7] for this class of nonsmooth functions first and second order conditions for local optimality (MIN). They are necessary and sufficient, eespectively. These generalizations of the classical KKT … Read more

    An Algorithm for Nonsmooth Optimization by Successive Piecewise Linearization

  • Sabrina Fiege
  • Andreas Griewank
  • Andrea Walther
    • Categories Nonsmooth Optimization Tags , , , ,

    We present an optimization method for Lipschitz continuous, piecewise smooth (PS) objective functions based on successive piecewise linearization. Since, in many realistic cases, nondifferentiabilities are caused by the occurrence of abs(), max(), and min(), we concentrate on these nonsmooth elemental functions. The method’s idea is to locate an optimum of a PS objective function by … Read more

    First and second order optimality conditions for piecewise smooth objective functions

  • Andreas Griewank
  • Andrea Walther
    • Categories Nonsmooth Optimization, Unconstrained Optimization Tags , , , , , , ,

    Any piecewise smooth function that is specified by an evaluation procedures involving smooth elemental functions and piecewise linear functions like min and max can be represented in the so-called abs-normal form. By an extension of algorithmic, or automatic differentiation, one can then compute certain first and second order derivative vectors and matrices that represent a … Read more