Maximization principles for lower semicontinuous functions.
A typical variational principle in optimization studies minimization of perturbations of a bounded from below lower semicontinuous function by elements from a prescribed family of functions (e.g., by linear continuous functions (Bishop-Phelps theorem, Stegall principle), by scalar modification of the norm (Ekeland variational principle, Borwein-Preiss variational principle), by continuous functions (Choban, Kenderov, Revalski) etc.). Symmetrically, one can study the same problem for maximization of a bounded from above upper semicontinuous function. In this talk we will study variational principles for maximization of perturbations of a lower semicontinuous function (not for minimization of such functions which is the usual case).
It turns out that the existence of a winning strategy for one of the players in the Banach-Mazur game played in the underlying space characterizes the validity of a generic variational principle for existence of solutions for the maximization of continuous bounded perturbations of a fixed lower semicontunuous function which is bounded from above. We give also some consequences of this result which give another (internal) characterizations for the underlying space in the case of the validity of the above principle. We also provide examples which outline the limits for having such results.