Generalized Convex Functions

Convexity is one of the most frequently used hypotheses in optimization theory. It is usually introduced to give global validity to propositions otherwise only locally true, for instance, a local minimum is also a global minimum for a convex function. Moreover, convexity is also used to obtain sufficiency for conditions that are only necessary, as with the classical Fermat theorem or with Kuhn-Tucker conditions in nonlinear programming. In microeconomics, convexity plays a fundamental role in general equilibrium theory and in duality theory. For more applications and historical reference, see, Arrow and Intriligator (1981), Guerraggio and Molho (2004), Islam and Craven (2005). The convexity of sets and the convexity and concavity of functions have been the object of many studies during the past one hundred years. Early contributions to convex analysis were made by Holder (1889), Jensen (1906), and Minkowski (1910, 1911). 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, convexity does not appear as a natural property of the various functions and domain encountered in such models. The property of convexity is invariant with respect to certain operations and transformations. However, for many problems encountered in economics and engineering the notion of convexity does no longer suffice. Hence, it is necessary to extend the notion of convexity to the notions of pseudo-convexity, quasi-convexity, etc. We should mention the early work by de de Finetti (1949), Fenchel (1953), Arrow and Enthoven (1961), Mangasarian (1965), Ponstein (1967), and Karamardian (1967). In the recent years, several extensions have been considered for the classical convexity. A significant generalization of convex functions is that of invex functions introduced by Hanson (1981). Hanson's initial result inspired a great deal of subsequent work which has greatly expanded the role and applications of invexity in nonlinear optimization and other branches of pure and applied sciences.

In this chapter, we shall discuss about various concepts of generalized convex functions introduced in the literature in last thirty years for the purpose of weakening the limitations of convexity in mathematical programming. Hanson (1981) introduced the concept of invexity as a generalization of convexity for scalar constrained optimization problems, and he showed that weak duality and sufficiency of the Kuhn-Tucker optimality conditions hold when invexity is required instead of the usual requirement of convexity of the functions involved in the problem.