On properties of geodesic semilocal E-preinvex functions

Abstract

The authors define a class of functions on Riemannian manifolds, which are called geodesic semilocal E-preinvex functions, as a generalization of geodesic semilocal E-convex and geodesic semi E-preinvex functions, and some of its properties are established. Furthermore, a nonlinear fractional multiobjective programming is considered, where the functions involved are geodesic E-η-semidifferentiability, sufficient optimality conditions are obtained. A dual is formulated and duality results are proved by using concepts of geodesic semilocal E-preinvex functions, geodesic pseudo-semilocal E-preinvex functions, and geodesic quasi-semilocal E-preinvex functions.

Generalized Geodesic Convexity on Riemannian Manifolds

“…The geodesic geodesic E-convexity was proposed on a Riemannian manifold in [20]. These results were further generalized in [21][22][23]. Recently, Ahmad et al [24] have discussed the geodesic sub-b-s convex functions and studied characterizations of these functions.…”

On the Characteristic Properties of Geodesic Sub-$ (\alpha,b,s )$-Preinvex

“…Hudzik and Maligrand in [8] studied two kinds of s-convexity (s ∈ (0, 1)) and sub-b-s-convex functions by using modulation s-convexity and sub-b-convexity, see [12]. Similarly, sub-b-s-pre invexity was presented the generalization for s-convex and b-preinvexity, see [11].…”

“…The concept of geodesic invexity in Riemannian manifold was introduced and preinvexity on a geodesic invex set were defined, further the relationship between geodesic invexity and preinvexity on manifolds was studied by Barani and Pouryayevali [4], while geodesic α-invexity and α-preinvex functions were defined in [2]. Further in the literature there are many more related generalizations of convexity, and new class of generalized convexity such as strongly α-invex and strongly geodesic α-preinvex functions etc., see [1,10,11].…”

Geodesic <inline-formula><tex-math id="M1">$ \mathcal{E} $</tex-math></inline-formula>-prequasi-invex function and its applications to non-linear programming problems

“…γ u1,u2 (ρ) ∈ S opt . Kilicman and Saleh [9] defined geodesic local E-invexity as follows: Definition 4.3. [9] A nonempty subset S of N is said to be geodesic local E-invex with respect to η :…”

“…Definition 4.4. [9] Let S be an open subset of N which is geodesic local E-invex with respect to η : N × N → T N . A function ψ : S → R is said to be geodesic local E-preinvex if for any u, v ∈ S, there exist ρ u,v ∈ (0, ρ u,v ] such that…”