The Contractibility of the Efficient Frontier of Three-Dimensional Simply-Shaded Sets ? J. Benoist† N. Popovici‡ Abstract The aim of this paper is to study the geometrical and topological structure of the efficient frontier of simply-shaded sets in the three-dimensional Euclidean space with respect to the usual positive cone. Our main result concerns the contractibility of the efficient frontier and refines a recent result of A. Daniilidis, N. Hadjisavvas and S. Schaible (1997) regarding the connectedness of the efficient outcome set for three-criteria optimization problems involving continuous semistrictly quasiconcave objective functions. Key words: vector optimization, efficiency, contractibility, semistrict quasiconcavity AMS subject classification: 90C29, 90C26 1 Introduction Among the topological properties of efficient sets in vector optimization, the connectedness was intensively studied in the last years under certain generalized convexity assumptions. However, even under more restrictive assumptions, in the literature there are only a few results on the contractibility of efficient sets, for which we refer the reader to the Luc's monograph on vector optimization (Ref. 1) and references therein. Recently, motivated by the practical importance of fractional programming, the connectedness of efficient sets for multicriteria optimization problems involving semistrictly quasiconcave objective ?This work was supported by a research grant of CNCSU under Contract Nr.
- actually quasiconcave
- z? ≥
- result concerns
- shading families
- problems involving
- vector optimization
- called simply-crossed
- simply