Convex, Quasiconvex, Pseudoconvex, Logarithmic Convex, αm-Convex, and Invex Functions

Abstract

The importance of convex functions is well known in optimization problems. Convex functions come up in many mathematical models used in economics, engineering, etc. More often than not, convexity appears as a natural property of the various functions and domains encountered in such models. Furthermore, the property of convexity is invariant with respect to certain operations and transformations. Numerous efficient theoretical and practical methods are available in the literature for determining the minimum of a convex function. But for many problems encountered in economics and engineering the notion of convexity does no longer suffice. Hence it was necessary to extend the notion of convexity into the notions of pseudoconvexity, quasiconvexity, etc. Such functions will be introduced and characterized in this chapter.