Partially non-convex minimax theorem and applications to remotal sets.

Abstract: Given a convex subset B  A of a locally convex space Y  X; and a function f : Y  X ! R such that B is compact and f (y; ); y 2 Y; are concave and upper semicontinuous, we establish in a Örst step a minimax inequality of the form maxy2B infx2A f (y; x)  infx2A supy2B0 f (y; x); where B0 is the set of points y 2 B such that f (y; ) is proper and convex.

The main di§erence with the classical minimax theorem is that, here, the set B0 does not need to be convex or compact. We use this result to give a new proof of the characterization of remotal sets, relying on the convexiy of the set of farthest points. We also propose new duals for inÖnite programming problems with zero-duality gap.

Posted on Oct 16, 2023 in Optimization and Equilibrium, Seminars