Separation of Two Sets in a Product Space

Nehse, Reinhard

doi:10.1002/mana.19800970115

Cited by 5 publications

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Nonvertical Affine Manifolds and Separation Theorems in Product Spaces

Thierfelder

1991

Mathematische Nachrichten

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In the paper we introduce a general separation concept for sets in product spaces X x 2, where 2 is partially ordered. Using some results about nonvertical affine manifolds we formulate some concrete separation theorems. Especially, we get assertions about the separation of sets by affine mappings, in which the order relation ''5'' is replaced by the relation ">". Thus, these assertions are suitable to characterize PARETO optimal solutions of vector optimization problems. IntroductionTo formulate optimality and duality assertions for classical nonlinear optimization problems (that means for problems in presence of functionals) the separation theorems of convex analysis give an excellent approach. In most cases the separation theorems are used in form of assertions about the existence of nonvertical support functionals of convex sets in product spaces X xIR ( X a real vector space, R the space of real numbers) or equivalently in form of HAHN-BANACH assertions as extension of linear functionals. To give a similar approach also for vector optimization problems the separation and extension theorems must be enlarged to product spaces X xZ, where 2 is a partially ordered vector space. Results of BONNICE/SILVERMAN [ 11, TO [ 141, VALA-DIER [15], BORWEIN [3], NEHSE [9], [lo] and other show, that a formal extension of these theorems is possible if and only if the order relation in 2 has the least upper bound property. However, all these results are riot suitable for using in vector optimization since by this approach we can characterize only strongly optimal solutions.The present paper is devoted to provide modificated separation theorems in product spaces in which analogously to NEHSE [ll] the order relation "5" (smaller or equal) is replaced by the relation ">" (not greater). By this, a t first the demand of comparability of the elements in Z is weakened. Second we can omit the least upper bound property in Z. Last not least with our separation concept we give a possibility to analyze PARETO optimal solutions of vector optimization problems.In detail, section 2 contains preliminary material. Section 3 is concerned with affine manifolds. We give some equivalent descriptions for nonverticality of such sets. I n

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