On Higher-Order Sensitivity Analysis in Nonsmooth Vector Optimization

Higher-order optimality conditions for set-valued optimization with ordering cones having empty interior using variational sets

“…The above corollary gives us other assumptions for necessary conditions of (P 1 ) and (P C ) in terms of radial-contingent derivatives proposed in Propositions 3.8(ii), 3.9 in [13].…”

“…Replacing the cone intC (B) in Propositions 3.10, 3.11 by a nonempty open cone Q (different from Y ), from Remark 3.12, we get immediately necessary conditions for Q-minimal solutions (see [4,13]) of (P 1 ) and (P 2 ) as follows. For (P 1 ), let (x 0 , y 0 ) ∈ gr(F • G).…”

“…Conditons similar to (3.3) were also used to obtain calculus rules of other generalized derivatives, for example, coderivatives, contingent derivatives, and radialcontingent derivatives in [13,22,24], resp.…”

“…These problems were also studied in terms of some kinds of generalized derivatives, such as radial sets and radial derivatives in [1,4], radial-contingent derivatives in [13], and contingent epiderivatives in [17]. Lemma 3.9 [19] Let F : X → 2 Y , (x 0 , y 0 ) ∈ gr F, and B be a base of C. If y 0 is a proper Henig efficient point of F(X ), then for some ∈ (0, δ),…”

“…To compare our results with those in [13], we recall the definition of the higherorder radial-contingent derivative of a mapping F : X → 2 Y at (x 0 , y 0 ) ∈ gr F, see Definition 3.1 in [13], as follows. n) : t n u n → 0, y 0 + t m n v n ∈ F(x 0 + t n u n )}.…”