Five Lectures on Differential Variational Inequalities

Abstract

Complex engineering and economic systems are often dynamic in nature and its time-varying variables are subject to algebraic restrictions that are cast in the form of equalities and inequalities; furthermore, these variables may be related to each other via some logical conditions. For such constrained dynamical systems, the classical approach based on ordinary differential equations (ODEs) alone is inadequate to fully capture the intricate details of the system evolution. Combining an ODE with a variational condition on an auxiliary algebraic variable that represents either a constrained optimization problem in finite dimensions or its generalization of a variational inequality, the mathematical paradigm of differential variational inequalities (DVIs) was born. On one hand, a DVI addresses the need to extend the century-old ODE paradigm to incorporate such elements as algebraic inequalities, mode changes, and logical relations, and on the other hand, introduces a temporal element into a static optimization or equilibrium problems. The resulting DVI paradigm transforms the basic science of dynamical systems by leveraging the modeling power and computational advances of constrained optimization and its extension of a variational inequality, bringing the former (dynamical systems) to new heights that necessitate renewed analysis and novel solution methods and endowing the latter (optimization) with a novel perspective that is much needed for its sustained development. Introducing the subject of the DVI and giving a survey of its recent developments, this report is an expanded version of five lectures that the author gave at the CIME Summer School on Centralized and Distributed Multi-agent Optimization Models and Algorithms held in Cetraro, Italy, June 23–27, 2014.