lipschitz continuity

Convergence of Finite-Dimensional Approximations for Mixed-Integer Optimization with Differential Equations

  • Falk M. Hante
  • Martin Schmidt
    • Categories (Mixed) Integer Nonlinear Programming, Nonlinear Optimization, Systems governed by Differential Equations Optimization Tags , , , ,

    We consider a direct approach to solve mixed-integer nonlinear optimization problems with constraints depending on initial and terminal conditions of an ordinary differential equation. In order to obtain a finite-dimensional problem, the dynamics are approximated using discretization methods. In the framework of general one-step methods, we provide sufficient conditions for the convergence of this approach … Read more

    On Stable Piecewise Linearization and Generalized Algorithmic Differentiation

  • Andreas Griewank
    • Categories Nonlinear Optimization, Nonsmooth Optimization, Optimization Software Design Principles Tags , , , , , , , , , , , , ,

    It is shown how functions that are defined by evaluation programs involving the absolute value function (besides smooth elementals), can be approximated locally by piecewise-linear models in the style of algorithmic, or automatic, differentiation (AD). The model can be generated by a minor modification of standard AD tools and it is Lipschitz continuous with respect … Read more

    On smooth relaxations of obstacle sets

  • Oliver Stein
  • Paul Steuermann
    • Categories Constrained Nonlinear Optimization, Convex Optimization Tags , , , , ,

    We present and discuss a method to relax sets described by finitely many smooth convex inequality constraints by the level set of a single smooth convex inequality constraint. Based on error bounds and Lipschitz continuity, special attention is paid to the maximal approximation error and a guaranteed safety margin. Our results allow to safely avoid … Read more

    Generalized differentiation with positively homogeneous maps: Applications in set-valued analysis and metric regularity

  • C.H. Jeffrey Pang
    • Categories Convex and Nonsmooth Optimization Tags , , , ,

    We propose a new concept of generalized differentiation of set-valued maps that captures the first order information. This concept encompasses the standard notions of Frechet differentiability, strict differentiability, calmness and Lipschitz continuity in single-valued maps, and the Aubin property and Lipschitz continuity in set-valued maps. We present calculus rules, sharpen the relationship between the Aubin … Read more

    Fischer-Burmeister Complementarity Function on Euclidean Jordan Algebras

  • Lingchen Kong
  • Levent Tunçel
  • Naihua Xiu
    • Categories Complementarity and Variational Inequalities Tags , ,

    Recently, Gowda et al. [10] established the Fischer-Burmeister (FB) complementarity function (C-function) on Euclidean Jordan algebras. In this paper, we prove that FB C-function as well as the derivatives of the squared norm of FB C-function are Lipschitz continuous. Citation Research Report CORR 2007-17, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada, … Read more