Projected dynamical systems and variational inequalities with applications
type: Item
instance of: scholarly article
title: Projected dynamical systems and variational inequalities with applications
zbMATH Open document ID: 0865.90018
publication date: 1996-08-26T00:00:00Z
review text: The variational inequality problem is defined as follows: for a closed convex set \(K\subset R^k\) and a function \(F:K\to R^k\) determine \(x^*\in K\) such that \(\langle F(x^*)^t,x-x^*\rangle\geq 0\;\forall x\in K\), where \(\langle.,.\rangle\) denotes the inner product in \(R^k\). Under appropriate conditions, the above may be shown to be equivalent to solving a system of nonlinear equations, solving a complementarity problem, solving an optimization exercise or solving a fixed point problem for some projection map. These relationships between a wide variety of problems provides the setting for the enquiry into solutions to the variational inequality problem described above and referred to as VI\((F,K)\). These discussions are also used to introduce the notion of a projected dynamical system. Let \(K\) be as above; for any \(v\in R^k\) and \(x\in K\) define the projection of the vector \(v\) at \(x\) (with respect to \(K\)), \(\Pi_K(x,v)\) by \[ \Pi_K(x,v)=\lim_{\delta\to 0} {P_K(x+\delta v)-x\over \delta}, \] where \(P_K(x)=\text{argmin}_{z\in K}|x-z|\), and \(|\cdot|\) denotes the Euclidean norm. Consider the system of ordinary differential equations defined by \(\dot x=\Pi_K(x,-F(x))\), where \(F(x)\) is a vector field on \(K\). Since the right-hand side is discontinuous on the boundary of \(K\), the notion of solution to the differential equation needs to be defined: \(x:[0,\infty)\to K\) is a solution if it is absolutely continuous and \(\dot x=\Pi_K(x,-F(x))\) save on a set of Lebesgue measure zero. Given an ordinary differential equation on the above type, the projected dynamic system PDS\((F,K)\) is defined to be the map \(\Phi:K\times R\to K\), where \(\Phi(x,t)=\phi_x(t)\) satisfies \(\dot\phi_x(t)=\Pi_K(\phi_x(t),-F(x(t)))\), \(\phi_x(0)=x\). The connection between VI\((F,K)\) and PDS\((F,K)\) rests on the result that if \(K\) is a convex polyhedron then the equilibrium points of PDS\((F,K)\) coincide with VI\((F,K)\). The book is based on the links me…
reviewed by: Anjan Mukherji
Mathematics Subject Classification ID: 91B50
Mathematics Subject Classification ID: 91B62
Mathematics Subject Classification ID: 91D25
Mathematics Subject Classification ID: 90-02
zbMATH DE Number: 920061
zbMATH Keywords: stability analysis
zbMATH Keywords: variational inequality problem
zbMATH Keywords: projected dynamical system
zbMATH Keywords: computation of equilibrium
zbMATH Keywords: oligopolistic market equilibria
zbMATH Keywords: spatial network oligopoly models
zbMATH Keywords: traffic network equilibrium
author: Anna Nagurney
author: Ding Zhang
MaRDI profile type: Publication
published in: International Series in Operations Research & Management Science
description: scientific article; zbMATH DE number 920061