Daniilidis, A., De Bernardi, C. A., & Miglierina, E. (2025). ABB theorems: results and limitations in infinite dimensions. Journal of Optimization Theory and Applications, 207(2), Article 38. https://doi.org/10.1007/s10957-025-02797-z
We construct a weakly compact convex subset of ℓ2 with nonempty interior that has an isolated maximal element, with respect to the lattice order 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.
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Project title:
Unilateralität und Asymmetrie in der Variationsanalyse: P 36344N (FWF - Österr. Wissenschaftsfonds)
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