Connectedness of Cone Superefficient Point Sets in Locally Convex Topological Vector Spaces

On super efficiency in set-valued optimization

“…Let F be ic-D-convexlike. Then exactly one of the following statements is true: [9] . If closed convex cone D has a bounded base B, then intD * = B st .…”

“…If closed convex cone D has a bounded base B, then intD * = B st . Lemma 2.3 [9] . Let Y be a locally convex space, D ⊂ Y be a closed convex pointed cone with a bounded base B, then …”

“…have been developed in several papers under some generalized convexity assumptions: cone-convexlikeness, cone-subconvexlikeness, and near cone-subconvexlikeness [3][4][5][6][7][8] , among them, the near cone-subconvexlikeness is the most general notion. The definition of near cone-subconvexlikeness does not require any topological structure of the cones D and E, the nonemptiness of the interior of these cones must be satisfied when proving optimization results [4][5][6][7][8][9][10][11][12] . Observe that this requirement does not hold in many optimization problems.…”

Connectedness of Cone-Efficient Solution Set for Cone-Quasiconvex Multiobjective Programming in Locally Convex Spaces

“…Among the topological properties of these sets, connectedness is a basic one, as it provides a possibility of continuously moving from one efficient solution to any other along the efficient alternatives only and has a close relationship to the fixed point property that is a useful argument in the economic equilibrium theory, and so it is of great interest to as (see [10,16] and cited references there). There has appeared a vast literature on this topic (see [1][2][3][4][5][6][7][8][10][11][12][13]15,16]) since Naccache first proved in 1978 that the efficient outcome set is connected for closed convex and cone-compact feasible outcome (see [12]). …”

“…In infinite dimensional spaces, Luc introduced the concepts of cone-quasiconvex and conecontinuous and proved that the weakly cone-efficient solution set is connected for cone-continuous cone-quasiconvex criteria under a weakened assumption on the compactness of the upper level sets (see [11]); Helbig introduced a different concept of cone-quasiconvex and proved that the weakly cone efficient solution set is connected for continuously cone-quasiconvex mapping (see [6]); Gong proved the connectedness of the efficient solution set in convex vector optimization for set-valued mapping in normed spaces (see [5]); recently, Hu and Ling obtained the connectedness of a cone-superefficient point set under the condition that the set is cone-convex and weakly cone-compact (see [7]). Of course, the following problem is of considerable interest: Whether the cone-efficient solution set is connected when the objective mapping is cone-quasiconvex on a convex compact set in topological vector spaces.…”