Evolution variational inequalities with general costs

TL;DR

This paper extends the theory of gradient flows to general cost functions, establishing properties and convergence of evolution variational inequalities driven by diverse divergences.

Contribution

It introduces a generalized framework for EVIs with new convexity notions and proves convergence of splitting schemes for these flows.

Findings

Established stability and energy identities for generalized EVIs.

Proved that EVI flows are limits of splitting schemes under certain assumptions.

Extended gradient flow theory beyond metric spaces to include Bregman and entropic divergences.

Abstract

We extend the theory of gradient flows beyond metric spaces by studying evolution variational inequalities (EVIs) driven by general cost functions $c$, including Bregman and entropic transport divergences. We establish several properties of the resulting flows, including stability and energy identities. Using novel notions of convexity related to costs $c$, we prove that EVI flows are the limit of splitting schemes, providing assumptions for both implicit and explicit iterations.