A Bregman firmly nonexpansive proximal operator for baryconvex optimization
This work addresses a theoretical optimization problem for researchers in convex and nonconvex optimization, but it appears incremental as it builds on existing proximal operator frameworks.
The authors tackled the problem of generalizing the proximal operator for baryconvex optimization by introducing a new operator based on convex combinations with minimax coefficient updates, proving it is Bregman firmly nonexpansive and that its fixed points correspond to critical points of a nonconvex function, and deriving continuous flows.
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.