On the Polyak convexity principle and its application to variational analysis

TL;DR

This paper extends Polyak's convexity principle from Hilbert spaces to certain uniformly convex Banach spaces, enabling new variational analysis tools for constrained optimization in these settings.

Contribution

It generalizes Polyak's convexity result to a broader class of Banach spaces, facilitating advanced variational analysis in infinite-dimensional spaces.

Findings

Extension of Polyak's convexity principle to uniformly convex Banach spaces

Development of a variational principle for constrained optimization in these spaces

Exploration of additional variational consequences

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.