The convexity theory plays an important role in many aspects of mathematical programming. In recent years, in order to relax convexity assumption, various generalized convexity notions have been obtained. One of them is the concept of ) , ( r p B− invexity defined by T.Antczak [1], which extended the class of B − invex functions with respect toη and b and the classes of ) , ( r p invex function...

The present paper deals with the properties of geodesic η-preinvex functions and their relationships with η-invex functions and strictly geodesic η-preinvex functions. The geodesic η-pre-pseudo-invex and geodesic η-pre-quasi-invex functions on the geodesic invex set are introduced and some of their properties are discussed.

In this paper, a geodesic α-invex subset of a Riemannian manifold is introduced. Geodesic α-invex and α-preinvex functions on a geodesic α-invex set with respect to particular maps are also defined. Further, we study the relationships between geodesic α-invex and α-preinvex functions on Riemannian manifolds. Some results of a non smooth geodesic α-preinvex function are also discussed using prox...

In this paper, we introduce a new class of nonsmooth pseudo-invex and nonsmooth quasi-invex functions to non-smooth variational problems. By using these concepts, numbers of necessary and sufficient conditions are established for a nonsmooth variational problem wherein Clarke’s generalized gradient is used. Also, weak, strong and converse duality are established. Keywords—Variational problem; N...

In this paper we consider some classes of α-preinvex and α-invex functions. We study some properties of these classes of α-preinvex (α-invex) functions. In particular, we establish the equivalence among the α-preinvex functions, α-invex functions and αη-monotonicity of their differential under some suitable conditions.

The purpose of this paper is to derive some criteria for vectorial prequasi-invex type functions via Jensen type inequalities. It is shown that a Jensen type inequality is sufficient and necessary for a vector function to be prequasi-invex under the condition of lower levelclosedness, cone lower semicontinuity, cone upper semicontinuity and semistrict prequasi-invexity, respectively. Analogous ...

The concept of a geodesic invex subset of a Riemannian manifold is introduced. Geodesic invex and preinvex functions on a geodesic invex set with respect to particular maps are defined. The relation between geodesic invexity and preinvexity of functions on manifolds is studied. Using proximal subdifferential, certain results concerning extremum points of a non smooth geodesic preinvex function ...