On the connectedness of the set of weakly efficient points of a vector optimization problem in locally convex spaces

Connectedness of Efficient Solutions in Multiple Objective Combinatorial Optimization

“…Naccache (1978) established connectivity for more general problems with closed, convex and K-compact outcome spaces where K is a closed, convex and pointed cone. Helbig (1990) generalized this to locally convex spaces.…”

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

“…In this paper, using the concept of cone-quasiconvex in [6], we study the connectedness of the set of cone-efficient solutions for multiobjective programming in locally convex Hausdorff topological vector spaces. We prove that the cone-efficient solution set is connected for multiobjective programming defined by a continuous cone-quasiconvex mapping on a compact convex set of alternatives.…”

“…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.…”

Connectedness of super efficient solution sets for set-valued maps in Banach spaces

“…Among the topological properties of these sets, connectedness is of interest. Several authors have studied this topic for point-valued functions in infinite dimensional spaces, it can be found in Hu and Hu [7], Helbig [8], Luc [10], and Fu and Zhou [4]. Gong [-5-6] has studied the connectedness of efficient solution sets in conestrongly quasiconvex set-valued maps in topological vector lattice and in convex set-valued maps in normed spaces, respectively.…”