Invex Functions (The Smooth Case)

Usually, generalized convex functions have been introduced in order to weaken as much as possible the convexity requirements for results related to optimization theory (in particular, optimality conditions and duality results), to optimal control problems, to variational inequalities, etc. For instance, this is the motivation for employing pseudo-convex and quasi-convex functions in [142, 143]; [228] use convexlike functions to give a very general condition for minimax problems on compact sets. Some approaches to generate new classes of generalized convex functions have been to select a property of convex functions which is to be retained and then forming the wider class of functions having this property: both pseudo-convexity and quasi-convexity can be assigned to this perspective. Other generalizations have been obtained through altering the expressions in the definition of convexity, such as the arcwise convex functions in [8] and [9], the (h, ϕ)-convex function in [17], the (α, λ)-convex functions in [27], the semilocally generalized convex functions in [113], etc.