Oliver Stein

How to Solve a Semi-infinite Optimization Problem

  • Oliver Stein
    • Categories Semi-infinite Programming Tags , , , , , , , ,

    After an introduction to main ideas of semi-infinite optimization, this article surveys recent developments in theory and numerical methods for standard and generalized semi-infinite optimization problems. Particular attention is paid to connections with mathematical programs with complementarity constraints, lower level Wolfe duality, semi-smooth approaches, as well as branch and bound techniques in adaptive convexification procedures. … 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

    A lifting method for generalized semi-infinite programs based on lower level Wolfe duality

  • Moritz Diehl
  • Oliver Stein
  • Boris Houska
  • Paul Steuermann
    • Categories Complementarity and Variational Inequalities, Robust Optimization, Semi-infinite Programming Tags , , , ,

    This paper introduces novel numerical solution strategies for generalized semi-infinite optimization problems (GSIP), a class of mathematical optimization problems which occur naturally in the context of design centering problems, robust optimization problems, and many fields of engineering science. GSIPs can be regarded as bilevel optimization problems, where a parametric lower-level maximization problem has to be … 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

    Twice differentiable characterizations of convexity notions for functions on full dimensional convex sets

  • Oliver Stein
    • Categories Convex Optimization Tags , , , ,

    We derive $C^2-$characterizations for convex, strictly convex, as well as uniformly convex functions on full dimensional convex sets. In the cases of convex and uniformly convex functions this weakens the well-known openness assumption on the convex sets. We also show that, in a certain sense, the full dimensionality assumption cannot be weakened further. In the … Read more