Sufficient optimality conditions and duality theorems for set-valued optimization problem under generalized cone convexity

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

“…Das and Nahak [8]- [19] introduced the notion of ρ-cone convex set-valued maps. They established the sufficient KKT conditions and studied the duality results for various types of set-valued optimization problems under contingent epiderivative and ρ-cone convexity assumptions.…”

“…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…”

Set-valued optimization problems via second-order contingent epiderivative

“…Das and Nahak [5,6,7,8,9,10,11,12,13,14,15] introduced the notion of ρ-cone convex set-valued maps. They establish the sufficient KKT conditions and study the duality results for various types of set-valued optimization problems under contingent epiderivative and ρ-cone convexity assumptions.…”

“…Definition 3.1. [5,8] Let X, Y be real normed spaces, A be a nonempty convex subset of X, K be a solid pointed convex cone in Y , e ∈ int(K), and F : X → 2 Y be a set-valued map, with A ⊆ dom(F ). Then F is said to be ρ-K-convex with respect to e on A if there exists ρ ∈ R such that…”

Optimality Conditions for Set-Valued Minimax Programming Problems via Second-Order Contingent Epiderivative

“…For special case, when φ : R n × R m → R and g : R n → R p are single-valued maps, we have a minimax programming problem (Schmitendorf (1977) Das and Nahak (2014, 2015, 2016a,b, 2017a,b,c, 2020 introduced the notion of ρ-cone convexity for set-valued maps. They establish the KKT sufficient conditions and develop the duality results for various types of set-valued optimization problems under contingent epiderivative as well as ρ-cone convexity assumptions.…”