Christian Kanzow

Inexact Penalty Decomposition Methods for Optimization Problems with Geometric Constraints

  • Christian Kanzow
  • Matteo Lapucci
    • Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , , , , ,

    This paper provides a theoretical and numerical investigation of a penalty decomposition scheme for the solution of optimization problems with geometric constraints. In particular, we consider some  situations where parts of the constraints are nonconvex and complicated, like cardinality constraints, disjunctive programs, or matrix problems involving rank constraints. By a variable duplication and  decomposition strategy, … Read more

    Mathematical Programs with Cardinality Constraints: Reformulation by Complementarity-type Constraints and a Regularization Method

  • Oleg Burdakov
  • Christian Kanzow
  • Alexandra Schwartz
    • Categories Complementarity and Variational Inequalities, Global Optimization, Nonlinear Optimization Tags , , , , , , ,

    Optimization problems with cardinality constraints are very dicult mathematical programs which are typically solved by global techniques from discrete optimization. Here we introduce a mixed-integer formulation whose standard relaxation still has the same solutions (in the sense of global minima) as the underlying cardinality-constrained problem; the relation between the local minima is also discussed in … Read more

    Smoothness Properties of a Regularized Gap Function for Quasi-Variational Inequalities

  • Nadja Harms
  • Oliver Stein
  • Christian Kanzow
    • Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization, Game Theory Tags , , , , ,

    This article studies continuity and differentiability properties for a reformulation of a finite-dimensional quasi-variational inequality (QVI) problem using a regularized gap function approach. For a special class of QVIs, this gap function is continuously differentiable everywhere, in general, however, it has nondifferentiability points. We therefore take a closer look at these nondifferentiability points and show, … Read more

    Gradient consistency for integral-convolution smoothing functions

  • James V. Burke
  • Tim Hoheisel
  • Christian Kanzow
    • Categories Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , ,

    Chen and Mangasarian (1995) developed smoothing approximations to the plus function built on integral-convolution with density functions. X. Chen (2012) has recently picked up this idea constructing a large class of smoothing functions for nonsmooth minimization through composition with smooth mappings. In this paper, we generalize this idea by substituting the plus function for an … Read more

    On Differentiability Properties of Player Convex Generalized Nash Equilibrium Problems

  • Nadja Harms
  • Oliver Stein
  • Christian Kanzow
    • Categories Constrained Nonlinear Optimization, Game Theory, Nonsmooth Optimization Tags , , , , ,

    This article studies differentiability properties for a reformulation of a player convex generalized Nash equilibrium problem as a constrained and possibly nonsmooth minimization problem. By using several results from parametric optimization we show that, apart from exceptional cases, all locally minimal points of the reformulation are differentiability points of the objective function. This justifies a … Read more