9. Generalizations of Convex Functions: Quasiconvex, Strictly Quasiconvex, and Pseudoconvex Functions

Beginning with Chap. 4, we have continually used the concepts of convex and concave functions in deriving optimality conditions and duality relations. Since not all properties of convex and concave functions are needed in establishing some of the previous results, a more general type of function will also work. For example, some results need only that the set Λα={x|x∈Γ,θ(x)≦α} be convex, where Γ is a convex set in Rn , θ is a numerical function defined on Γ, and α is any real number. Now if θ is a convex function on Γ, the convexity of Λα is assured by Theorem 4.1.10. However, θ need not be convex in order that Λα be convex. A function θ which is quasi-convex on Γ has this property [Nikaidô 54]. Another property of differentiable convex functions that was used in previous results was this: If ∇θ(x¯)(x−x¯)≧0 , then θ(x)≧θ(x¯) . This property follows from 6.1.1, and has the obvious consequence that if ∇θ(x¯)=0 , then θ(x)≧θ(x¯) . Not only convex functions have this property. Pseudoconvex functions [Tuy 64, Mangasarian 65], to be introduced in this chapter, also have this property. By using only the properties of convex functions that are needed in establishing some of the previous results, these results are extended to a larger class of functions.