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Support separation theorems and their applications to vector surrogate reverse duality. (English) Zbl 0770.49004
Separation theorems play a crucial role in optimization theory. One of the best known separation theorems is the following: Given a nonempty closed convex subset \(A\) of a Hausdorff locally convex space \(X\) and a point \(x\in X\backslash A\), there always exists a nonzero continuous functional \(\varphi\in X^*\) such that (1) \(\varphi(x)\geq\sup\varphi(A)\). The author considers the question when the inequality in (1) could be replaced with equality. This problem was first studied by I. Singer. He has proved the existence of such a functional under the assumptions that the space \(X\) is a normable linear space and \(A\) is a bounded set.
In this paper a satisfying answer to this question for the general case when \(X\) is a Hausdorff locally convex space and no constraining assumptions are imposed is given. The obtained results are then used to establish some strong duality principles concerning the surrogate reverse duality in vector optimization.
MSC:
49J27 Existence theories for problems in abstract spaces
49N15 Duality theory (optimization)
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References:
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