Set-valued fractional programming problems under generalized cone convexity

Sufficiency and duality of set-valued fractional programming problems via second-order contingent epiderivative

“…Definition 3.1. [8,11] Let A be a nonempty convex subset of R n , e ∈ int(R m + ) and F : R n → 2 R m be a set-valued map, with A ⊆ dom(F ). Then F is said to be ρ-R m + -convex with respect to e on A if there exists ρ ∈ R such that…”

“…Das and Nahak [11] constructed an example of ρ-cone convex set-valued map, which is not cone convex. They also characterized ρ-cone convexity of set-valued maps in terms of contingent epiderivative.…”

“…They also formulated the higher-order Mond-Weir dual for set-valued optimization problems and studied the duality theorems under convexity assumptions. In 2015, Das and Nahak [11] established the sufficient Karush-Kuhn-Tucker (KKT) optimality conditions for set-valued fractional programming problems under contingent epiderivative and ρ-cone convexity assumptions. They also studied the duality results of parametric, Mond-Weir, Wolfe, and mixed types for the problem.…”

Optimization Problems With Difference of Set-Valued Maps Under Generalized Cone Convexity

“…We introduce the notion of ρ-cone convexity of set-valued maps in [6]. For ρ = 0, we have the usual notion of cone convexity of set-valued maps.…”

“…If ρ > 0, then F is said to be strongly ρ-cone convex, for ρ = 0 we get the usual notion of cone convexity and if ρ < 0, then F is said to be weakly ρ-cone convex. In [6], we give an example of ρ-cone convex set-valued map, which is not cone convex.…”