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Connectedness of approximate efficient solutions for generalized semi-infinite vector optimization problems. (English) Zbl 1430.90514

This paper deals with sufficient conditions for the connectedness of the set of approximate efficient solutions for the generalized semi-infinite vector optimization problem with set-valued objective mappings (GSIVOP for short). After giving some preliminaries, a class of GSIVOP’s is introduced in Section 2.1. Then some definitions related to generalized convexity are summarized in Section 2.2. Some basic facts on upper semi-continuous and lower semi-continuous mappings are presented in Section 2.3. Section 3 establishes sufficient conditions under which the constraint set mapping for the problem GSIVOP is convex valued and upper semi-continuous. Section 4 deals with the connectedness of approximate efficient solutions of the problem GSIVOP. First, in Section 4.1, a scalarized semi-infinite programming problem is introduced and some properties of this problem are presented. Section 4.2 gives sufficient conditions for the connectedness of the set of approximate efficient points (Theorem 4.6) and the set of approximate efficient solutions (Theorem 4.7) of the problem GSIVOP. The results of this paper generalize the results from X. H. Gong [J. Optim. Theory Appl. 83, No. 1, 83–96 (1994; Zbl 0845.90104)] and Z.-F. Li and S.-Y. Wang [Math. Methods Oper. Res. 48, No. 2, 207–217 (1998; Zbl 0928.90081)].

MSC:

90C29 Multi-objective and goal programming
90C31 Sensitivity, stability, parametric optimization
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