Elements of Convex Analysis

Abstract

The purpose of this chapter is to provide some notions and fundamental results of convex analysis which will be used throughout this book. Starting with the notion of convexity, some propositions on convex and lower semi-continuous functionals as well as on the minimization of functionals on convex sets are given. The notion of subdifferential is introduced and its relation to one-sided Gâteaux-differentiability is illustrated. There follows the definition of the conjugate functional and some propositions on the conjugacy operation. The chapter closes with some elements of the theory of maximal monotone operators. Our attention is concentrated on the maximal monotone operators on ℝ as they allow a compact formulation of general classes of variational inequalities. In the present chapter, the relation between convex analysis and the theory of variational inequalities becomes clear.KeywordsVariational InequalityConvex SubsetMonotone OperatorMaximal MonotoneConvex AnalysisThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.