Chapter

Handbook of Generalized Convexity and Generalized Monotonicity

Volume 76 of the series Nonconvex Optimization and Its Applications pp 3-87

# Introduction to Convex and Quasiconvex Analysis

• Johannes B.G. FrenkAffiliated withDepartment of Economics, Econometric Institute, Erasmus University
• , Gábor KassayAffiliated withFaculty of Mathematics and Computer Science, Babes Bolyai University

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## Abstract

In the first chapter of this book the basic results within convex and quasiconvex analysis are presented. In Section 2 we consider in detail the algebraic and topological properties of convex sets within ℝn together with their primal and dual representations. In Section 3 we apply the results for convex sets to convex and quasiconvex functions and show how these results can be used to give primal and dual representations of the functions considered in this field. As such, most of the results are well known with the exception of Subsection 3.4 dealing with dual representations of quasiconvex functions. In Section 3 we consider applications of convex analysis to noncooperative game and minimax theory, Lagrangian duality in optimization and the properties of positively homogeneous evenly quasiconvex functions. Among these result an elementary proof of the well-known Sion’s minimax theorem concerning quasiconvex-quasiconcave bifunctions is presented, thereby avoiding the less elementary fixed point arguments. Most of the results are proved in detail and the authors have tried to make these proofs as transparent as possible. Remember that convex analysis deals with the study of convex cones and convex sets and these objects are generalizations of linear subspaces and affine sets, thereby extending the field of linear algebra. Although some of the proofs are technical, it is possible to give a clear geometrical interpretation of the main ideas of convex analysis. Finally in Section 5 we list a short and probably incomplete overview on the history of convex and quasiconvex analysis.

### Keywords

Convex Analysis Quasiconvex Analysis Noncooperative games Minimax Optimization theory