Convex Analysis and Subdifferential Calculus

Abstract

This chapter deals with several properties of convex functions, especially in connection with their regularity, on the one hand, and the characterization of their minimizers, on the other. We shall explore sufficient conditions for a convex function to be continuous, as well as several connections between convexity and differentiability. Next, we present the notion of subgradient, a generalization of the concept of derivative for nondifferentiable convex functions that will allow us to characterize their minimizers. After discussing conditions that guarantee their existence, we present the basic (yet subtle) calculus rules, along with their remarkable consequences. Other important theoretical and practical tools, such as the Fenchel conjugate and the Lagrange multipliers, will also be studied. These are particularly useful for solving constrained problems.