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A new ABB theorem in Banach spaces. (English) Zbl 0949.90083
The paper is concerned with the Arrow-Barankin-Blackwell theorem proved by K. J. Arrow, E. W. Barankin, and D. Blackwell [Contrib. Theory of Games. II, Ann. Math. Studies 28, 87-91 (1953; Zbl 0050.14203)], which states that the minimal points, relative to the componentwise partial ordering in \(\mathbb{R}^n\), of a closed and convex set \(S\) are limits of sequences of minimum points of strictly positive functionals. This result has been extended into more general ones, some of which can be found in the author’s previous paper [J. Math. Anal. Appl. 158, 47-54 (1991; Zbl 0734.49008)] with normed space setting. The author has proved a new version of the theorem in a Banach space \(X\) equipped with a partial ordering induced by a closed, convex and pointed cone \(C\); that is, if \(C\) is supposed to have a weakly compact base, the minimal points of a closed and convex set \(S\) in \(X\) are in the strong closure of the minimum points of the linear continuous functionals which are strictly positive in \(C\setminus\{0\}\). The author provides two proofs of the theorem; the first proof is based on the celebrated drop theorem and the second proof is based on a method introduced by X. H. Gong [J. Optimization Theory Appl. 83, 83-96 (1994; Zbl 0845.90104)].

MSC:
90C29 Multi-objective and goal programming
90C48 Programming in abstract spaces
46B40 Ordered normed spaces
49J99 Existence theories in calculus of variations and optimal control
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
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