Strong Pseudoconvexity and Strong Quasiconvexity of Non-differentiable Functions

In this chapter, we introduce the concept of strong pseudomonotonicity and strong quasimonotonicity of set-valued maps of higher order. Non-differentiable strong pseudoconvex/quasiconvex functions of higher order are characterized by the strong pseudomonotonicity/quasimonotonicity of their corresponding set-valued maps. As a by-product, we solve the open problem (converse part of Proposition 6.2) of Karamardian and Schaible (J. Optim. Theory Appl. 66:37–46, 1990) for the more general case as strong pseudoconvexity for non-smooth, locally Lipschitz continuous functions.