An extension of the Polyak convexity principle with application to nonconvex optimization

TL;DR

This paper extends Polyak's convexity principle to uniformly convex sets in Banach spaces, providing conditions under which nonlinear transformations preserve convexity, with applications to nonconvex optimization problems.

Contribution

It introduces a quantitative condition linking convexity modulus, regularity, and derivative Lipschitz continuity to ensure convexity preservation under nonlinear maps.

Findings

Established a condition for convexity preservation in Banach spaces.

Applied the principle to solution existence and characterization in nonconvex optimization.

Connected convexity preservation to Lagrangian duality in constrained problems.

Abstract

The main problem considered in the present paper is to single out classes of convex sets, whose convexity property is preserved under nonlinear smooth transformations. Extending an approach due to B.T. Polyak, the present study focusses on the class of uniformly convex subsets of Banach spaces. As a main result, a quantitative condition linking the modulus of convexity of such kind of set, the regularity behaviour around a point of a nonlinear mapping and the Lipschitz continuity of its derivative is established, which ensures the images of uniformly convex sets to remain uniformly convex. Applications of the resulting convexity principle to the existence of solutions, their characterization and to the Lagrangian duality theory in constrained nonconvex optimization are then discussed.