On the Polyak convexity principle and its application to variational analysis

2013/03/29 by Uderzo, Amos
#49J52 #52A05 #90C46 #90C48 #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1303.7443

Abstract

According to a result due to B.T. Polyak, a mapping between Hilbert spaces, which is C^1,1 around a regular point, carries a ball centered at that point to a convex set, provided that the radius of the ball is small enough. The present paper considers the extension of such result to mappings defined on a certain subclass of uniformly convex Banach spaces. This enables one to extend to such setting a variational principle for constrained optimization problems, already observed in finite dimension, that establishes a convex behaviour for proper localizations of them. Further variational consequences are explored.

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