Variational Inequalities Problem

Abstract

This course support is dedicated to the Variational Inequality Problem (V IP ), de- signed for Masterís 2 students in Optimization and Optimal Control option. Variational inequalities provide a powerful, uniÖed mathematical framework that generalizes and encompasses a wide range of fundamental problems in applied math- ematics, economics, and engineering. This includes optimization problems (both constrained and unconstrained), complementarity problems, Öxed-point problems, and Nash equilibrium problems in game theory. Therefore, knowing how to formu- late and solve (V IP ) is a crucial skill for handling complex real-world systems that involve optimality and equilibrium. This document is structured to guide the student from the foundational principles to the forefront of numerical techniques. Chapter One establishes the core concepts, introducing the formal deÖnition of the variational inequality problem, its geometric interpretation, and its profound con- nections with optimization and other mathematical problems. Chapter Two delves into the theory behind (V IP ). We will classify di§erent types of (V IP ) based on the properties of the underlying mapping (e.g., monotonicity, strong monotonicity) and rigorously examine the crucial questions of existence and uniqueness of solutions, which form the base of any numerical approach. Chapter Three marks the transition into computational methods. We will explore the family of projection methods, the foundational algorithms for solving (V IP ). The student will learn about their derivation and their convergence properties. Chapter Four builds upon this foundation by focusing on the advanced and highly e¢ cient class of projection and contraction methods. These algorithms, which en- hance basic projection steps with specially designed contraction steps, o§er robust convergence without restrictive assumptions. By the end of this course, the student will has gained a comprehensive understanding of both the theory of variational inequalities and the numerical tools essential for solving them, providing a strong foundation for his research and professional work in optimization and optimal control.