無限連立線形不等式に対するGordanの定理

概要

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It is well-known that Gordan's lemma, one of the theorems of the alternatives, gives a necessary and sufficient condition for the existence of solutions to finite systems of strict linear inequalities. The aim of this paper is to show that Gordan's lemma can be extended for the infinite case if we adopt a suitable non-Archimedean structure as the domain of solutions to infinite systems of strict linear inequalities. For this purpose we introduce an extended structure of the field of real numbers IR with both an infinitely large number w and an infinitely small number ε, showing that this structure coincides with the set of lexicographically ordered vectors. The main result is derived as a consequence of the lexicographical separation theorem of Hausner and Wendel, Klee, Martinez-Legaz and Singer that any two disjoint convex sets in IR^n can be separated lexicographically.
つくば国際大学の論文
2004-00-00