Parallel Computation of Variational Inequalities and Projected Dynamical Systems with Applications

Abstract

This chapter presents parallel algorithms, along with numerical results, for the computation of solutions to finite-dimensional variational inequality problems and for projected dynamical systems. The variational inequality problem has been used to formulate a plethora of equilibrium problems in diverse applications, whereas the theory of projected dynamical systems has been introduced only recently to allow for the study of dynamic phenomena, subject to constraints. The connection between these two problems is notable in that the set of solutions to a variational inequality problem coincides with the set of solutions to a projected dynamical system. Hence, problems that have, heretofore, been studied principally in the static setting of equilibrium states can now be addressed through their disequilibrium behavior. Applications that can be studied using these companion methodologies range from congested urban transportation systems and spatial price equilibrium problems to general financial equilibrium problems. Such problems are naturally complex and large-scale and, hence, parallel computation can play a pivotal role in their analysis and solution. In this chapter we address precisely these questions.Key wordsVariational inequalitiesprojected dynamical systemsparallel computationlarge-scale equilibrium problemsdisequilibrium behaviorapplications to operations researcheconomicsfinance