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A generalization of Dubovitskij-Milyutin theorem. (English) Zbl 0999.49005

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 \(K\).)
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”.
We give a generalization of the theorem, which is symmetric and works even in cases where none of the cones is solid”.

MSC:

49J27 Existence theories for problems in abstract spaces
49J53 Set-valued and variational analysis
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
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