Maximal Monotone Maps and Variational Inequalities
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Monotone maps are of fundamental importance in optimization and variational inequalities. A particularly important special class of monotone maps are those that are maximal monotone and that share several properties similar to those of subdifferentials of convex functions. In a certain way they naturally generalize the concept of subdifferentials and provide a unified framework to formulate convex optimization problems and more generally saddle points problem that arise in several important applications such as game theory and equilibrium problems in economy. This general framework permits us not only to recast these problems in the format of solving generalized equations, but also leads to the development of methods for their solutions. The purpose of this chapter is to give a self-contained introduction to the theory of maximal monotone maps, which are useful for analyzing and solving variational inequalities.