chapter  4
32 Pages

Convexity, Generalized Convexity, and Applications

Department of Product and Systems Design, University of the Aegean, Hermoupolis, Greece

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 4.3 Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.4 Quasiconvex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.5 Pseudoconvex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.6 On the Minima of Generalized Convex Functions . . . . . . . . . . . . . . . 161 4.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

4.7.1 Sufficiency of the KKT Conditions . . . . . . . . . . . . . . . . . . . . . . 163 4.7.2 Applications in Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

4.8 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

The study of convex functions has given rise to one of the most beautiful branches of mathematical analysis, namely convex analysis, which has had a considerable influence on other more abstract areas, such as functional analysis. At the same time, convex functions have been applied to model many problems in engineering, economics, management, etc.