ABB Theorems: Results and Limitations in Infinite Dimensions

Aris Daniilidis (), Carlo Alberto Bernardi () and Enrico Miglierina ()
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Aris Daniilidis: TU Wien
Carlo Alberto Bernardi: Università Cattolica del Sacro Cuore
Enrico Miglierina: Università Cattolica del Sacro Cuore

Journal of Optimization Theory and Applications, 2025, vol. 207, issue 2, No 19, 13 pages

Abstract: Abstract We construct a weakly compact convex subset of $$\ell ^{2}$$ ℓ 2 with nonempty interior that has an isolated maximal element, with respect to the lattice order $$\ell _{+}^{2}$$ ℓ + 2 . Moreover, the maximal point cannot be supported by any strictly positive functional, which shows that the Arrow-Barankin-Blackwell theorem fails. This example discloses the pertinence of the assumption that the cone has a bounded base for the validity of the result in infinite dimensions. Under this latter assumption, the equivalence of the notions of strict maximality and maximality is established.

Keywords: ABB theorem; Efficient point; Positive functional; Density; 90C29; 46B20; 46N10 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10957-025-02797-z

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