Handbook of Generalized Convexity and Generalized Monotonicity

Abstract

Convexity of functions plays a central role in many branches of ap- plied mathematics. One of the reasons is that it is very well fitted to extremum problems. For instance, some necessary conditions for the existence of a minimum become also sufficient in the presence of con- vexity. However, by far not all real-life problems can be described by a convex mathematical model. In many cases, nonconvex functions pro- vide a much more accurate representation of reality. It turns out that often these functions, in spite of being nonconvex, retain some of the nice properties and characteristics of convex functions. For instance, their presence may ensure that necessary conditions for a minimum are also sufficient or that a local minimum is also a global one. This led to the introduction of several generalizations of the classical concept of a convex function. It happened in various fields such as economics, en- gineering, management science, probability theory and various applied sciences for example, mostly during the second half of the last (20th) century.