Arcwise cone-quasiconvex multicriteria optimization
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 be a convex cone in a real linear space and consider the preorder relation defined for all by
Definition 1 We will say that a function , defined on a nonempty subset of a topological space , is arcwise -quasiconvex if the level set is arcwise connected for every point . Actually, is arcwise -quasiconvex if for all and such that and there exists a continuous function such that
Multicriteria optimization problems
Consider a vector function , defined on a nonempty set . If is a nonempty set of indices, the notation will stand for a function , where denotes the cardinality of and the indices are implicitly given by and . In particular, we have . With each nonempty selection of indices, , we associate an optimization problem:
Notice that is a scalar optimization problem if is a
Acknowledgement
Nicolae Popovici’s research was supported by the grant CNCSIS ID 2261 under Contract 543/2008.
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Cited by (10)
Decomposition of generalized vector variational inequalities
2012, Nonlinear Analysis, Theory, Methods and ApplicationsCitation Excerpt :Motivated by the facility location theory, this question has been investigated for the first time by Lowe et al. in [2], where it was shown that the set of weakly efficient solutions of a convex multicriteria optimization problem actually is the union of the sets of efficient solutions of all associated subproblems. Extensions of this result to different classes of generalized convex multicriteria optimization problems were obtained by Malivert and Boissard [3], Popovici [4,5], and by La Torre and Popovici [6]. The principal aim of our paper is to develop a similar decomposition approach for a special class of generalized vector variational inequalities introduced by Lee et al. in [7], which can be seen as a combination of multicriteria optimization problems and Stampacchia-type vector variational inequalities.
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