A Bregman firmly nonexpansive proximal operator for baryconvex optimization

TL;DR

This paper introduces a generalized proximal operator based on convex combinations of objectives, demonstrating its Bregman firm nonexpansiveness and analyzing its fixed points and continuous flows in a nonconvex setting.

Contribution

It proposes a novel Bregman firmly nonexpansive proximal operator for baryconvex optimization, extending classical proximal methods to nonconvex functions with combined geometries.

Findings

The operator is Bregman firmly nonexpansive with respect to a combined divergence.

Fixed points correspond to critical points of a nonconvex function.

Derived continuous flows related to the operator.

Abstract

We present a generalization of the proximal operator defined through a convex combination of convex objectives, where the coefficients are updated in a minimax fashion. We prove that this new operator is Bregman firmly nonexpansive with respect to a Bregman divergence that combines Euclidean and information geometries; and that its fixed points are given by the critical points of a certain nonconvex function. Finally, we derive the associated continuous flows.