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Zbl 0999.49005
Averbuch, V.
A generalization of Dubovitskij-Milyutin theorem. (English)
[J] Acta Univ. Carol., Math. Phys. 41, No.2, 5-6 (2000). ISSN 0001-7140

From the text: ``One of the fundamental theorems of convex analysis is the Dubovitskij-Milyutin theorem. Let $X$ be a Hausdorff locally convex space, let $K_1,\dots, K_n$ be convex cones in $X$ (with the vertex at $0$), all but one open, and let the intersection of all $n$ cones be empty. Then there exist elements $x^*_1\in K^*_1,\dots, x^*_n\in K^*_n$, not all zero, such that $x^*_1+\cdots+ x^*_n= 0$. (Here $K^*$ denotes the dual (polar) cone to $K$.)\par This theorem is evidently nonsymmetric: one of the cones stands by itself. Besides it is supposed in the theorem that almost all cones are ``solid''.\par We give a generalization of the theorem, which is symmetric and works even in cases where none of the cones is solid''.

MSC 2000:: *49J27 Optimal control problems in abstract spaces (existence)
49J53 Set-valued and variational analysis
46N10 Appl. of functional analysis in optimization and math. programming

Keywords: Dubovitskij-Milyutin theorem; symmetry; Hausdorff locally convex space; convex cones

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Zentralblatt MATH Berlin [Germany]