Projection based algorithms for variational inequalities

This dissertation is about the theory and iterative algorithms for solving variational inequalities. Chapter 1 introduces the problem, various situations in which variational inequalities arise naturally, reformulations of the problem, several characteristics of the problem based on those reformulations, as well as the basic existence and uniqueness results. Following that, chapter 2 describes the general approaches to solving variational inequalities, focusing on projection based methods towards the end, with some convergence results. That chapter also discusses the merits and demerits of those approaches. In chapter 3, we describe a relaxed projection method, and a descent method for solving variational inequalities with some examples. An application of the descent framework to a game theory problem leads to an algorithm for solving box constrained variational inequalities. Relaxed projection methods require a sequence of parameters that approach zero, which leads to slow convergence as the iterates approach a solution. Chapter 4 describes a local convergence result that can be used as a guideline for finding a bound on the parameter as a relaxed projection algorithm reaches a solution

A von Neumann Alternating Method for Finding Common Solutions to Variational Inequalities

Modifying von Neumann's alternating projections algorithm, we obtain an alternating method for solving the recently introduced Common Solutions to Variational Inequalities Problem (CSVIP). For simplicity, we mainly confine our attention to the two-set CSVIP, which entails finding common solutions to two unrelated variational inequalities in Hilbert space.Comment: Nonlinear Analysis Series A: Theory, Methods & Applications, accepted for publicatio