generalized equations

Variational Convergence of Bifunctions: Motivating Applications

  • Alejandro Jofré
  • Roger J.B. Wets
    • Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization, Game Theory Tags , , , , , , , , ,

    It’s shown that a number of variational problems can be cast as finding the maxinf-points (or minsup-points) of bivariate functions, coveniently abbreviated to bifunctions. These variational problems include: linear and nonlinear complementarity problems, fixed points, variational inequalities, inclusions, non-cooperative games, Walras and Nash equilibrium problems. One can then appeal to the theory of lopsided convergence … Read more

    From convergence principles to stability and optimality conditions

  • Diethard Klatte
  • Alexander Y. Kruger
  • Bernd Kummer
    • Categories Nonsmooth Optimization Tags , , , , ,

    We show in a rather general setting that Hoelder and Lipschitz stability properties of solutions to variational problems can be characterized by convergence of more or less abstract iteration schemes. Depending on the principle of convergence, new and intrinsic stability conditions can be derived. Our most abstract models are (multi-) functions on complete metric spaces. … Read more

    Implicit Multifunction Theorems in complete metric spaces

  • Van Ngai Huynh
  • Huu Tron Nguyen
  • Michel Thera
    • Categories Infinite Dimensional Optimization, Nonsmooth Optimization Tags , , , ,

    In this paper, we establish some new characterizations of the metric regularity of implicit multifunctions in complete metric spaces by using the lower semicontinuous envelopes of the distance functions for set-valued mappings. Through these new characterizations it is possible to investigate implicit multifunction theorems based on coderivatives and on contingent derivatives as well as the … Read more

    Real-Time Optimization as a Generalized Equation

  • Mihai Anitescu
  • Victor M. Zavala
    • Categories Complementarity and Variational Inequalities, Control Applications, Nonlinear Optimization Tags , , , ,

    We establish results for the problem of tracking a time-dependent manifold arising in online nonlinear programming by casting this as a generalized equation. We demonstrate that if points along a solution manifold are consistently strongly regular, it is possible to track the manifold approximately by solving a linear complementarity problem (LCP) at each time step. … Read more