Convex Functions

Hoang Tuy
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Hoang Tuy: Institute of Mathematics

Chapter Chapter 2 in Convex Analysis and Global Optimization, 2016, pp 39-86 from Springer

Abstract: Abstract This chapter presents the basic concepts and facts about convex functions. After standard definitions, we discuss the basic operations that preserve convexity, including a theorem on the concavity of the geometric mean of m concave positive functions. Theorems on consistent and inconsistent systems of convex inequalities are then established, to provide the foundation for the study of differential properties of convex functions, subdifferential calculus, and conjugate functions. Further, extremal properties of convex functions are discussed, in particular necessary and sufficient conditions for the minimizer or maximizer of a convex function over a convex set. Especially, an important place is given to the proof of the classic minimax theorem in its strongest version together with its application to the theory of optimality conditions and Lagrange duality for convex and generalized convex optimization, including conic optimization and semidefinite programming.

Date: 2016
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DOI: 10.1007/978-3-319-31484-6_2

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