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Contractibility of the efficient frontier of three dimensional simply shaded sets

25 pages
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


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1
The Contractibility of the Efficient Frontier of
Three-Dimensional Simply-Shaded
J. Benoist
Abstract
N. Popovici
Sets
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
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. 46174 LACO, UPRESSA 6090, University of Limoges, France. Faculty of Mathematics and Computer Science, Babes-Bolyai University of Cluj, Romania.
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functions was investigated by Daniilidis, Hadjisavvas and Schaible in Ref. 2 for the three-criteria
case and extended by Benoist in Ref. 3 for the multicriteria case.
The aim of this paper is to revisit the three criteria case from the contractibility point of view,
by exploring the geometric structure of the orthogonal projection of the efficient frontier on a
suitable plane. In fact, this approach allows us to establish a more general result concerning the
contractibility of some abstract sets in the Euclidean plane, induced by a so-called simply-crossed shading family, which consists of some halflines satisfying simple geometric properties.
The paper is organized as follows. In Section 2 we recall some preliminary notations and
definitions in vector optimization and we introduce the notion of three-dimensional simply-shaded
set. Section 3 is devoted to the axiomatization of simply-crossed shading families in the plane and to
present some basic results, which play a key role in the sequel. In Section 4 we state our main result
on the contractibility of the nucleus of a simply-crossed shading family. Section 5 is devoted to
establish the relationship between simply-shaded sets and simply-crossed shading families, in order
to get the contractibility of the efficient frontier of simply-shaded sets. In particular, we obtain
the contractibility of the efficient outcome in three-criteria semistrictly quasiconcave optimization
problems. In Section 6 we conclude by pointing out some interesting open problems.
2 Simply-shaded sets in the Euclidean space Our study here will be limited to the three-dimensional Euclidean spaceR3, partially ordered by the positive coneR+3some notions and results included in the sections 2 and 3 may be extended, but to higher dimensions.
We denote byand>the usual component-wise order relations defined inR3by xy⇐⇒xyR3+andx > y⇐⇒xyR3+\ {0}For every setYR3, we denote by MaxYits efficient frontier, i.e. MaxY={yY:Y(y+R3+) ={y}} It is well-known that for any setYR3we have MaxY (= MaxYR+3) which shows that for studying the topological structure of the efficient frontier of a setYit causes no loss of generality to
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