Stochastic Variational Inequality Problems: Applications, Analysis, and Algorithms

Abstract

Deterministic finite-dimensional variational inequality problems represent a tool for capturing a broad range of optimization and equilibrium problems in application set- tings ranging from traffic equilibrium models and energy markets to communication networks, among others. Increasingly, these settings are complicated by risk and uncer- tainty, and although deterministic models represent an important first step, a compre- hensive examination of the stochastic generalization of such problems is in order. One possible model for capturing uncertainty in this realm is a finite-dimensional stochastic variational inequality problem, which is defined by a deterministic set and a map whose components contain expectations. Unfortunately, traditional analytical and compu- tational techniques largely fail to extend naturally in resolving this problem when the expectations are defined over a general probability space. This tutorial intends to provide a review of a subset of the analytical and algorithmic advances that have emerged in the resolution of this problem. We motivate the study of the stochastic variational inequality problem by considering two applications, the first drawn from strategic bidding in power markets and the second arising in the design of cognitive radio systems. This discussion paves the way for the following sets of contributions of this tutorial and concludes by demonstrating the utility of the presented tools on one of the application settings. (i) First, existence statements for the stochastic varia- tional inequality problem are complicated by the need to evaluate a multidimensional integral. By combining Lebesgue convergence theorems with existence statements for finite-dimensional analogs, a set of integration-free existence statements are provided for the stochastic variational inequality problem. Furthermore, these statements are extended to the regime where the integrands of the expectation are multivalued maps arising, for instance, from a stochastic nonsmooth Nash game. (ii) Second, when the expectations are unavailable as closed-form expressions, deterministic schemes can- not easily resolve stochastic variational problems, prompting the need for leveraging Monte Carlo sampling schemes. One such avenue lies in the use of stochastic approx- imation schemes. The earliest among these relied on the strong monotonicity of the map. Hybrid variants of such schemes that combine Tikhonov regularization and proximal-point methods are also suggested, both of which weaken the need for strong monotonicity of the map and are characterized by desirable almost-sure convergence properties. A shortcoming of the earlier schemes is the limited guidance provided for the choice of step-length sequences. This concern is partially resolved in the presen- tation of a self-tuned step-length stochastic approximation scheme that prescribed a step-length rule that adapts to problem parameters such as Lipschitz and monotonic- ity constants.