Second-Order Optimality Conditions with the Envelope-Like Effect for Set-Valued Optimization

Second-order efficient optimality conditions for set-valued vector optimization in terms of asymptotic contingent epiderivatives

“…In most known necessary conditions, (F, G) and its derivatives are used, for example in [11,13,26,27,46]. Inspired by the idea in [28,30], the Aubin property is employed to obtain a sharper second-order necessary conditions involving separately derivatives of F and G. From there, constraint qualifications of the Kurcyusz-Robinson-Zowe type, not qualification condition in terms of (F, G), can be invoked to get Karush-Kuhn-Tucker multiplier rules for problem (P).…”

“…In this case, Theorem 4.3 improves the corresponding result of Theorem 4.1 in [46], since the authors of [46] used the derivatives of a disjunction map, composed from the objective and the constraints, and the regularity assumptions involving these maps. (iii) Note that, the authors in [28] use the cone-Aubin properties of both the objective and the constraint maps to separate the derivatives of them. In our works, we only assume this property for only the objective map or the constraint map.…”

“…We can check that the second-order contingent derivative and epiderivative of F at (x 0 , y 0 ) does not exist, then Theorem 3.1 in [19] and Theorems 3.1 and 3.2 in [28] cannot be employed. Furthermore, F is not C-convex, hence Theorem 4.1 in [46] cannot also be used.…”

Second-order efficiency conditions for $C^{1,1}$-vector equilibrium problems in terms of contingent derivatives and applications

“…Definition 2.1. [1,6,16] Let M be a subset of X and let x, u ∈ X. (i) The contingent cone (resp., adjacent cone and interior tangent cone) of M at x is defined as…”

“…They used the first-order Bouligand tangent cone and the second-order tangent cone to a set at an optimal point given to obtain the optimality conditions for weak efficient solutions of problem CP. Recently, by using the concepts of second-order contingent derivatives, second-order asymptotic contingent derivatives and second-order composed contingent derivatives, Khanh and Tung [16] received the Karush-Kuhn-Tucker second-order optimality conditions for nonsmooth set-valued optimization problems with attention to the envelope-like effect, Su [19,20] obtained second-order optimality conditions for vector equilibrium problems in terms of contingent derivatives and epiderivatives with stable functions, and Luu [14] established second-order necessary optimality conditions for nonsmooth vector equilibrium problems using the Palés-Zeidan type second-order directional derivatives.…”