chapter 9, Additional Generalizations of Concavity

Excerpt

In this chapter we discuss some additional families of generalized concave functions. These families are considerably broader than those presented in the previous chapters.

First we discuss a generalization of concavity via support properties, due to Ben-Tal and Ben-Israel (1976). A well-known property that may serve to define a concave function over an open convex set is that its graph can be supported from above at each point by a hyperplane. This property is fundamental in the derivation of optimality, duality, and in the construction of algorithms for concave programming. A natural generalization of concavity is obtained by considering functions whose graphs are supported from above by supports that are not necessarily linear. This leads to the concept of F-concave functions, which is the subject of Section 9.1. We shall discuss properties of these functions and present several examples. One example ties F-concavity to a classical generalization of concavity developed by Beckenbach (1937); another example shows that the family of F-concave functions can become extremely large.

Further, in Section 9.2, following the work of Ortega and Rheinboldt (1970), and Avriel and Zang (1980), we present extensions of the families of generalized concave functions introduced in Chapter 3. The idea underlying the definitions of the new families is to examine the behavior of the function under consideration along continuous arcs instead of line segments as is done with the definitions of the ordinary classes of generalized concave functions.