Nonsmooth Invexities, Invariant Monotonicities and Nonsmooth Vector Variational-Like Inequalities with Applications to Vector Optimization

Abstract

It is well known that the convexity of functions plays a vital role in mathematical economics, engineering, management, optimization theory, etc. This concept in linear spaces relies on the possibility of connecting any two points of the space by the line segment between them. Since convexity is often not enjoyed the real problems, several classes of functions have been defined and studied for the purpose of weakening the limitations of convexity. In 1981, Hanson [33] realized that the convexity requirement, utilized to prove sufficient optimality conditions for a differentiable mathematical programming problem, can be further weakened by substituting the linear term y − x appearing in the definition of differentiable convex, pseudoconvex and quasiconvex functions with an arbitrary vector-valued function. In view of this idea, Hanson [33] (see also Craven [12]) introduced the concept of invexity by replacing the linear term y − x in the definition of convex function by a vector-valued function η(y, x).