Higher-Order Variational Sets and Higher-Order Optimality Conditions for Proper Efficiency in Set-Valued Nonsmooth Vector Optimization

First and second-order optimality conditions without differentiability in multivalued vector optimization

“…In order to establish sufficient conditions, certain relaxed convexity assumptions are usually to be imposed on objective mappings and constraints. For example, in [27,30] C-star-shapedness and pseudoconvexity are used along with variational sets. In [29], using the weak Clarke epiderivative, cone semilocal convexlikeness assumptions are imposed.…”

Optimality and duality in set-valued optimization using higher-order radial derivatives

“…Several kinds of higher-order derivatives have been developed for set-valued mappings by different authors. To some extent, higherorder derivatives in the related literatures can be divided into two categories: First, the existences of higherorder derivatives depend on the choice of lower-order directions, for instance, higher-oder contingent (adjacent) derivatives [1], the generalized higher-order contingent (adjacent) derivatives [2], cone-directed higher-order contingent (adjacent) derivatives [3], higher-order generalized contingent (adjacent) epiderivatives [4], higher-order weak epiderivatives [5], and variational sets [6,7,8], etc. ; Second, the direction of higher-order derivatives does not depend on lower-order direction, for example, Higher-order Studniarski derivative [9,10,11] enjoys this advantage.…”

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

“…In [18,19], Khanh and Tuan proposed variational sets of type 1 and type 2, known as another kind of generalized derivatives, and their applications to optimality conditions in set-valued optimization. These sets were defined as follows Definition 1.1 [18] (i) The mth-order variational set of type 1 of the mapping F at (x 0 , y 0 ) ∈ gr F is (ii) The mth-order variational set of type 2 of the mapping F at (x 0 , y 0 ) ∈ gr F is W m (F, x 0 , y 0 , v 1 , .…”

“…, v m−1 ) (ii) The variational sets are bigger than the image set of the contingent derivative since we allow the flexibility x F − → x 0 in the definition. Hence, necessary optimality conditions obtained in separations are stronger than many known ones, see [18,19]. (iii) The main difference between the variational sets of type 1 and type 2 is that V m has a local character in both the domain and the image of F (since x n → x 0 and t n → 0 + appear in the definition), while W m has this character only in the domain (h n > 0 occurs in the image of F).…”

“…Realizing advantages of variational sets in some topics, see [2,18,19], and observing that their calculus rules and applications were obtained only for variational sets of type 1, see [3], we aim to develop calculus rules of variational sets of type 2. Next, we establish higher-order optimality conditions in set-valued optimization when the ordering cones have empty interior.…”