Finitely determined functions and convex optimization

Abstract : 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.