Complementarity and Variational Inequalities

Local Convergence of the Method of Multipliers for Variational and Optimization Problems under the Sole Noncriticality Assumption

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

    We present local convergence analysis of the method of multipliers for equality-constrained variational problems (in the special case of optimization, also called the augmented Lagrangian method) under the sole assumption that the dual starting point is close to a noncritical Lagrange multiplier (which is weaker than second-order sufficiency). Local superlinear convergence is established under the … Read more

    A hybrid proximal extragradient self-concordant primal barrier method for monotone variational inequalities

  • Renato D.C. Monteiro
  • Benar Fux Svaiter
  • Mauricio Romero
    • Categories Complementarity and Variational Inequalities Tags ,

    In this paper we present a primal interior-point hybrid proximal extragradient (HPE) method for solving a monotone variational inequality over a closed convex set endowed with a self-concordant barrier and whose underlying map has Lipschitz continuous derivative. In contrast to the method of “R. D. C. Monteiro and B. F. Svaiter, Iteration-Complexity of a Newton … Read more

    A game-theoretic approach to computation offloading in mobile cloud computing

  • Valeria Cardellini
  • Francisco Facchinei
  • Veronica Piccialli
  • Vittoria De Nitto Persone'
  • Valerio Di Valerio
  • Vincenzo Grassi
  • Francesco Lo Presti
    • Categories Applications - OR and Management Sciences, Complementarity and Variational Inequalities, Game Theory

    We consider a three-tier architecture for mobile and pervasive computing scenarios, consisting of a local tier of mobile nodes, a middle tier (cloudlets) of nearby computing nodes, typically located at the mobile nodes access points but characterized by a limited amount of resources, and a remote tier of distant cloud servers, which have practically infinite … Read more

    Inverse Parametric Optimization with an Application to Hybrid System Control

  • Paul J. Goulart
  • Andreas B. Hempel
  • John Lygeros
    • Categories Complementarity and Variational Inequalities, Control Applications, Convex Optimization Tags ,

    We present a number of results on inverse parametric optimization and its application to hybrid system control. We show that any function that can be written as the difference of two convex functions can also be written as a linear mapping of the solution to a convex parametric optimization problem. We exploit these results in … Read more

    A direct splitting method for nonsmooth variational inequalities

  • Yunier Bello-Cruz
  • R. Diaz Millan
    • Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization, Nonsmooth Optimization

    We propose a direct splitting method for solving nonsmooth variational inequality problems in Hilbert spaces. The weak convergence is established, when the operator is the sum of two point-to-set and monotone operators. The proposed method is a natural extension of the incremental subgradient method for nondifferentiable optimization, which explores strongly the structure of the operator … Read more

    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