Complementarity and Variational Inequalities – Page 18

Variational Convergence of Bifunctions: Motivating Applications

Published: 2011/02/04

Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization, Game Theory fixed points, generalized equations, inclusion systems, inequality systems, ky fan functions, lop-convergence, lopsided convergence, maxinf-points, nash and walras equilibrium points, variational inequality

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

On the Dynamic Stability of Electricity Markets

Published: 2011/01/31, Updated: 2011/02/09

Mihai Anitescu

Victor M. Zavala

Complementarity and Variational Inequalities, Dynamic Programming, Game Theory dynamics markets, efficiency, electricity, game theory, stability

In this work, we present new insights into the dynamic stability of electricity markets. In particular, we discuss how short forecast horizons, incomplete gaming, and physical ramping constraints can give rise to stability issues. Using basic concepts of market efficiency, Lyapunov stability, and predictive control, we construct a new stabilizing market design. A numerical case … Read more

A Continuous Dynamical Newton-Like Approach to Solving Monotone Inclusions

Published: 2010/12/15

Hédy Attouch

Benar Fux Svaiter

Complementarity and Variational Inequalities, Infinite Dimensional Optimization, Systems governed by Differential Equations Optimization absolutely continuous trajectories, dissipative dynamical systems, levenberg-marquardt algorithm, lyapunov analysis, maximal monotone operators, newton-like algorithm, non-autonomous differential equations, numerical convex optimization, weak asymptotic convergence

We introduce non-autonomous continuous dynamical systems which are linked to Newton and Levenberg-Marquardt methods. They aim at solving inclusions governed by maximal monotone operators in Hilbert spaces. Relying on Minty representation of maximal monotone operators as lipschitzian manifolds, we show that these dynamics can be formulated as first-order in time differential systems, which are relevant … Read more

A Parallel Inertial Proximal Optimization Method

Published: 2010/11/17, Updated: 2011/07/01

Jean-Christophe Pesquet

Nelly Pustelnik

Complementarity and Variational Inequalities, Convex Optimization, Parallel Algorithms augmented lagrangian method, convex optimization, douglas-rachford algorithm, maximal monotone operators, parallel methods, proximal methods

The Douglas-Rachford algorithm is a popular iterative method for finding a zero of a sum of two maximal monotone operators defined on a Hilbert space. In this paper, we propose an extension of this algorithm including inertia parameters and develop parallel versions to deal with the case of a sum of an arbitrary number of … Read more

First order optimality conditions for mathematical programs with semidefinite cone complementarity constraints

Published: 2010/11/16

Chao Ding

Defeng Sun

Jane J. Ye

Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization, Linear, Cone and Semidefinite Programming c-stationary conditions, constraint qualifications, m-stationary conditions, mathematical program with semidefinite cone complementarity constraints, necessary optimality conditions, s-stationary conditions

In this paper we consider a mathematical program with semidefinite cone complementarity constraints (SDCMPCC). Such a problem is a matrix analogue of the mathematical program with (vector) complementarity constraints (MPCC) and includes MPCC as a special case. We derive explicit expressions for the strong-, Mordukhovich- and Clarke- (S-, M- and C-)stationary conditions and give constraint … Read more

The Split Variational Inequality Problem

Published: 2010/09/20

Yair Censor

Aviv Gibali

Simeon Reich

Complementarity and Variational Inequalities, Constrained Nonlinear Optimization constrained vip, product space, split minimization, split variational inequality problem (svip), split zeros

We propose a new variational problem which we call the Split Variational Inequality Problem (SVIP). It entails finding a solution of one Variational Inequality Problem (VIP), the image of which under a given bounded linear transformation is a solution of another VIP. We construct iterative algorithms that solve such problems, under reasonable conditions, in Hilbert … Read more

Iteration-complexity of block-decomposition algorithms and the alternating minimization augmented Lagrangian method

Published: 2010/08/22

Renato D.C. Monteiro

Benar Fux Svaiter

Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization alternating direction method, augmented lagrangian method, block-decomposition, complexity, convex optimization, monotone inclusion, splitting methods

In this paper, we consider the monotone inclusion problem consisting of the sum of a continuous monotone map and a point-to-set maximal monotone operator with a separable two-block structure and introduce a framework of block-decomposition prox-type algorithms for solving it which allows for each one of the single-block proximal subproblems to be solved in an … Read more

Two stage stochastic equilibrium problems with equilibrium constraints: modeling and numerical schemes

Published: 2010/08/14

Huifu Xu

Dali Zhang

Complementarity and Variational Inequalities, Game Theory, Stochastic Programming convergence analysis, nash stationary point, sample average approximation, stochastic equilibrium programs with equilibrium constraints

This paper presents a two stage stochastic equilibrium problem with equilibrium constraints(SEPEC) model. Some source problems which motivate the model are discussed. Monte Carlo sampling method is applied to solve the SEPEC. The convergence analysis on the statistical estimators of Nash equilibria and Nash stationary points are presented. ArticleDownload View PDF

Approximating Stationary Points of Stochastic Mathematical Programs with Equilibrium Constraints via Sample Averaging

Published: 2010/08/11, Updated: 2010/08/25

Huifu Xu

Jane J. Ye

Complementarity and Variational Inequalities, Nonsmooth Optimization, Stochastic Programming c-stationary point, coderivative, graphical convergence, m-stationary point, sample average approximation, smpec

We investigate sample average approximation of a general class of one-stage stochastic mathematical programs with equilibrium constraints. By using graphical convergence of unbounded set-valued mappings, we demonstrate almost sure convergence of a sequence of stationary points of sample average approximation problems to their true counterparts as the sample size increases. In particular we show the … Read more

Optimality conditions for various efficient solutions involving coderivatives: from set-valued optimization problems to set-valued equilibrium problems

Published: 2010/08/02, Updated: 2017/04/24

Xuan Duc Ha Truong

Complementarity and Variational Inequalities, Multi-Criteria Optimization, Nonsmooth Optimization coderivative, necessary optimality conditions, normal cone, pareto efficient solutions, properly efficient solutions, quasi weakly efficient solutions, relative efficient solutions, set-valued equilibrium problem, set-valued optimization problem, strongly efficient solutions, sufficient optimality conditions, vector equilibrium problem, vector variational inequality, weakly efficient solutions

We present a new approach to the study of a set-valued equilibrium problem (for short, SEP) through the study of a set-valued optimization problem with a geometric constraint (for short, SOP) based on an equivalence between solutions of these problems. As illustrations, we adapt to SEP enhanced notions of relative Pareto efficient solutions introduced in … Read more