Complementarity and Variational Inequalities

An approximation scheme for a class of risk-averse stochastic equilibrium problems

  • Juan Pablo Luna
  • Claudia Sagastizábal
  • Mikhail V. Solodov
    • Categories Complementarity and Variational Inequalities, Stochastic Programming Tags , , , , ,

    We consider two models for stochastic equilibrium: one based on the variational equilibrium of a generalized Nash game, and the other on the mixed complementarity formulation. Each agent in the market solves a one-stage risk-averse optimization problem with both here-and-now (investment) variables and (production) wait-and-see variables. A shared constraint couples almost surely the wait-and-see decisions … Read more

    Regularizing Bilevel Nonlinear Programs by Lifting

  • Hans Georg Bock
  • Sven Leyffer
  • Kathrin Hatz
  • Johannes P. Schlöder
    • Categories Complementarity and Variational Inequalities, Nonlinear Optimization Tags , , , ,

    This paper considers a bilevel nonlinear program (NLP) whose lower-level problem satisfies a linear independence constraint qualification (LICQ) and a strong second-order condition (SSOC). One would expect the resulting mathematical program with complementarity constraints (MPCC), whose constraints are the first-order optimality conditions of the lower-level NLP, to satisfy an MPEC-LICQ. We provide an example which … Read more

    Nonlinear Equilibrium for optimal resource allocation

  • Roman Polyak
    • Categories Complementarity and Variational Inequalities, Game Theory Tags , , , , ,

    We consider Nonlinear Equilibrium (NE) for optimal allocation of limited resources. The NE is a generalization of the Walras-Wald equilibrium, which is equivalent to J. Nash equilibrium in an n-person concave game. Finding NE is equivalent to solving a variational inequality (VI) with a monotone and smooth operator on $\Omega = \Re_+^n\cross\Re_+^m$. The projection on … Read more

    GENERALIZATIONS OF THE DENNIS-MOR\’E THEOREM II

  • Asen L. Dontchev
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , ,

    This paper is a continuation of our previous paper were we presented generalizations of the Dennis-Mor\’e theorem to characterize q-superliner convergences of quasi-Newton methods for solving equations and variational inequalities in Banach spaces. Here we prove Dennis-Mor\’e type theorems for inexact quasi-Newton methods applied to variational inequalities in finite dimensions. We first consider variational inequalities … Read more

    A double projection method for solving variational inequalities without monotonicity

  • Ye Minglu
  • He Yiran
    • Categories Complementarity and Variational Inequalities Tags , ,

    We present a double projection algorithm for solving variational inequalities without monotonicity. If the solution of dual variational inequality does exist, then the sequence produced by our method is globally convergent to a solution. Under the same assumption, the sequence produced by known methods has only a subsequence converging to a solution. Numerical experiments are … 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

    The s-Monotone Index Selection Rule for Criss-Cross Algorithms of Linear Complementarity Problems

  • Nagy Adrienn
  • Zsolt Csizmadia
  • Tibor Illés
    • Categories Complementarity and Variational Inequalities, Linear, Cone and Semidefinite Programming Tags , , ,

    In this paper we introduce the s-monotone index selection rules for the well-known crisscross method for solving the linear complementarity problem (LCP). Most LCP solution methods require a priori information about the properties of the input matrix. One of the most general matrix properties often required for finiteness of the pivot algorithms (or polynomial complexity … Read more

    A New Error Bound Result for Generalized Nash Equilibrium Problems and its Algorithmic Application

  • Axel Dreves
  • Francisco Facchinei
  • Andreas Fischer
  • Markus Herrich
    • Categories Complementarity and Variational Inequalities, Game Theory, Nonlinear Optimization Tags , , , , ,

    We present a new algorithm for the solution of Generalized Nash Equilibrium Problems. This hybrid method combines the robustness of a potential reduction algorithm and the local quadratic convergence rate of the LP-Newton method. We base our local convergence theory on an error bound and provide a new sufficient condition for it to hold that … Read more

    Abstract Newtonian Frameworks and Their Applications

  • Alexey F. Izmailov
  • A. S. Kurennoy
    • Categories Complementarity and Variational Inequalities, Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , , , , ,

    We unify and extend some Newtonian iterative frameworks developed earlier in the literature, which results in a collection of convenient tools for local convergence analysis of various algorithms under various sets of assumptions including strong metric regularity, semistability, or upper-Lipschizt stability, the latter allowing for nonisolated solutions. These abstract schemes are further applied for deriving … Read more

    AN EFFICIENT ALGORITHM FOR SECOND-ORDER CONE LINEAR COMPLEMENTARITY PROBLEMS

  • Weihong Yang
  • Lei-Hong Zhang
    • Categories Complementarity and Variational Inequalities, Second-Order Cone Programming Tags , , , ,

    Recently, the globally uniquely solvable (GUS) property of the linear transformation $M\in R^{n\times n}$ in the second-order cone linear complementarity problem (SOCLCP) receives much attention and has been studied substantially. Yang and Yuan [30] contributed a new characterization of the GUS property of the linear transformation, which is formulated by basic linear-algebra-related properties. In this … Read more