m-stationarity

A General Penalty-Method and a General Regularization-Method for Cardinality-Constrained Optimization Problems

  • Felix P. Broesamle
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , , , ,

    We consider cardinality-constrained optimization problems (CCOPs), which are general nonlinear programs with an additional constraint limiting the number of nonzero continuous variables. The continuous reformulation of CCOPs involves complementarity constraints, which pose significant theoretical and computational challenges. To address these difficulties, we propose and analyze two numerical solution approaches: a general penalty method and a … Read more

    Relaxation methods for pessimistic bilevel optimization

  • Alain Zemkoho
    • Categories Bilevel Optimization, Complementarity and Variational Inequalities, Nonsmooth Optimization Tags , , , ,

    We consider a smooth pessimistic bilevel optimization problem, where the lower-level problem is convex and satisfies the Slater constraint qualification. These assumptions ensure that the Karush-Kuhn-Tucker (KKT) reformulation of our problem is well-defined. We then introduce and study the (i) Scholtes, (ii) Lin and Fukushima, (iii) Kadrani, Dussault and Benchakroun, (iv) Steffensen and Ulbrich, and … 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

    Elastic-Mode Algorithms for Mathematical Programs with Equilibrium Constraints: Global Convergence and Stationarity Properties

  • Mihai Anitescu
  • Paul Tseng
  • Stephen Wright
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization Tags , , , , , ,

    The elastic-mode formulation of the problem of minimizing a nonlinear function subject to equilibrium constraints has appealing local properties in that, for a finite value of the penalty parameter, local solutions satisfying first- and second-order necessary optimality conditions for the original problem are also first- and second-order points of the elastic-mode formulation. Here we study … Read more