Characterizations of super efficiency in cone-convexlike vector optimization with set-valued maps

Henig Efficiency in Vector Optimization with Nearly Cone-subconvexlike Set-valued Functions

“…Corollary 3.1 generalizes and unifies the recent results of Mehra (see [18,Theorem 4.1), and Rong and Wu (see [21,Theorem 5.1). Especially, Corollary 3.1 generalizes in the following several aspects.…”

“…As an interesting application of the results in this paper, we establish Lagrange multiplier theorems for super efficiency in vector optimization with nearly cone-subconvexlike set-valued function. This generalizes and unifies the recent results of Mehra (see [18], Theorems 4.1 and 4.2), and Rong and Wu (see [21,Theorem 5.1), etc.…”

“…Many results have been obtained. For instance, Li [16] extended the concept of cone-subconvexlikeness of single-valued functions to set-valued functions and characterized the Benson proper efficiency in terms of scalarization, Lagrange multiplier and duality under the assumption of cone-subconvexlike; Mehra [18] , and Rong and Wu [21] studied super efficiency in vector optimization with cone-convexlike set-valued functions and nearly cone-convexlike set-valued functions, respectively; Yang [24] introduced nearly cone-subconvexlike set-valued function, a Lagrangian multiplier theorem and two scalarization theorems are established; Gong [10] gave the scalarization results to the Henig efficient points set and presented some optimality conditions for Henig efficient solution pair of set-valued vector optimization problem under the assumption of cone-convexlike; Qiu and Fu [20] discussed the connectedness of super efficient solution sets in vector optimization with cone-convexlike setvalued functions. However, it is well known that, without a bounded base for the ordering cone, super efficient points may not exist even for a very well behaved compact set (see [5], example 4).…”

Strong duality with super efficiency in set-valued optimization

“…Later, Zheng [23] extended the concept of super efficiency from normed spaces to locally convex topological vector spaces. Some achievements related to super efficiency in vector optimization were developed in terms of scalarization, Lagrange multiplier, Lagrange duality, saddle point, topological property, etc., see [3,6,9,10,12,13,[19][20][21]. Recently, Zheng et al [24] and Bao and Mordukhovich [1] dealt with super efficiency in Banach spaces by employing tools of variational analysis.…”

Tightly Proper Efficiency in Vector Optimization with Nearly Cone-Subconvexlike Set-Valued Maps

“…Recently, several authors have turned their interests to vector optimization of setvalued maps, for instance, see [13][14][15][16][17][18] . Gong 19 discussed set-valued constrained vector optimization problems under the constraint ordering cone with empty interior.…”