Arcwise cone-quasiconvex multicriteria optimization

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Abstract

The aim of this paper is to study the structure of the efficient sets within a special class of multicriteria optimization problems, involving objective functions which are continuous and arcwise quasiconvex with respect to a certain family of cones. In particular, it is shown that such a problem is Pareto reducible and its efficient outcome set is strongly contractible.

Introduction

The role played by the generalized convexity in optimization is nowadays well recognized. In this paper we introduce a new concept of quasiconvexity for vector functions, which is shown to have some interesting applications in multicriteria optimization, especially as regards the Pareto reducibility and the strong contractibility of the efficient outcome set.

The notion of Pareto reducibility, introduced by Popovici in [8], allows one to reduce the complexity of a multicriteria problem by considering new problems obtained from the original one by selecting a certain number of criteria. More precisely, a multicriteria optimization problem is said to be Pareto reducible if its weakly efficient solutions actually are efficient solutions for the problem itself or for a subproblem obtained from it by selecting certain criteria. In [9] some results concerning the Pareto reducibility and the contractibility of efficient sets have been established for multicriteria optimization problems involving lexicographic quasiconvex objective functions. Our paper aims to extend these results.

Section snippets

Arcwise cone-quasiconvex functions

Let C be a convex cone in a real linear space Y and consider the preorder relation defined for all (y,y)Y×Y by yCy:yy+C.

Definition 1

We will say that a function f:DY, defined on a nonempty subset D of a topological space X, is arcwise C-quasiconvex if the level set f1(yC){xDf(x)yC}={xDf(x)Cy} is arcwise connected for every point yY. Actually, f is arcwise C-quasiconvex if for all yY and x1,x2D such that f(x1)Cy and f(x2)Cy there exists a continuous function γ:[0,1]D such that γ(0)=x1

Multicriteria optimization problems

Consider a vector function f=(f1,,fn):DRn, defined on a nonempty set D. If IIn is a nonempty set of indices, the notation fI will stand for a function fI(fi1,,fik):DRk, where k|I| denotes the cardinality of I and the indices i1,,ik are implicitly given by I={i1,,ik} and i1<<ik. In particular, we have fIn=f. With each nonempty selection of indices, IIn, we associate an optimization problem: (PI){MinimizefI(x)subject toxD.

Notice that (PI) is a scalar optimization problem if I is a

Acknowledgement

Nicolae Popovici’s research was supported by the grant CNCSIS ID 2261 under Contract 543/2008.

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