A nonlinear version of the Hahn-Banach theorem and some applications

M. Ruiz Galán

The existence of a linear functional that is bounded above by a given sublinear functional in a real vector space is guaranteed by the Hahn—Banach theorem. We first replace the role of the sublinear functional with a certain convex function whose effective domain has nonempty algebraic interior (not necessarily containing the null vector) and satisfies a condition on some of its slopes, and determine the compactness for an adequate topology of the set of the corresponding bounded above linear functionals. Then, we state a Hahn—Banach-type result by characterizing the existence of a function in a pointwise compact topological space of functions and satisfying a boundedness condition, which extends the Hahn—Banach theorem to a nonlinear framework. As a consequence, we derive some results in optimization, as a nonlinear Gordan’s theorem of the alternative.

Keywords: Hahn—Banach theorem, theorems of the alternative

GT11-2 MA-2 Continuous Optimization. Tribute to Marco Antonio López
September 5, 2019  4:05 PM
I2L7. Georgina Blanes building

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M. J. Gisbert Francés, M. J. Cánovas Cánovas, R. Henrion, J. Parra López

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