Finitely determined functions and convex optimization

Mohammed Bachir (1, 2) , Adrien Fabre (2, 3, 4) , Sebastian Tapia-Garcia (5)

1 Equations d'evolution
2 UP1 - Université Paris 1 Panthéon-Sorbonne
3 PSE - Paris School of Economics
4 PJSE - Paris Jourdan Sciences Economiques
5 DIM - Departamento de Ingeniera Matematica [Santiago]

We study the notion of {\it finitely determined functions} defined on a topological vector space $E$ equipped with a biorthogonal system. This notion will be used to obtain a necessary and sufficient condition for a convex function to attain a minimum at some point. An application to the Karush-Kuhn-Tucker theorem will be given. For real-valued convex functions defined on a Banach space with a Schauder basis, the notion of finitely determined function coincides with the classical continuity but outside the convex case there are many finitely determined nowhere continuous functions.

Mots clés: Finitely deter-mined function; Schauder basis; Convex optimization; Karush-Kuhn-Tucker theorem msc: 46N10; Directional derivatives

Domaines: Analyse fonctionnelle [math.FA]

Mohammed Bachir, Adrien Fabre, Sebastian Tapia-Garcia. Finitely determined functions and convex optimization. 2019. ⟨hal-01713851v2⟩