A partial differential equation for the strictly quasiconvex envelope

TL;DR

This paper introduces a new PDE regularization for quasiconvex functions that ensures well-posedness and allows for numerical approximation of the strictly quasiconvex envelope, resulting in smoother solutions.

Contribution

It proposes a stronger PDE regularization for quasiconvex functions, enabling convergence and numerical computation of the strictly quasiconvex envelope.

Findings

Solutions are strictly convex and smoother than previous regularizations.

Finite difference schemes converge to the quasiconvex envelope.

The new PDE regularization improves numerical stability and approximation quality.

Abstract

In a series of papers Barron, Goebel, and Jensen studied Partial Differential Equations (PDE)s for quasiconvex (QC) functions \cite{barron2012functions, barron2012quasiconvex,barron2013quasiconvex,barron2013uniqueness}. To overcome the lack of uniqueness for the QC PDE, they introduced a regularization: a PDE for $\e$-robust QC functions, which is well-posed. Building on this work, we introduce a stronger regularization which is amenable to numerical approximation. We build convergent finite difference approximations, comparing the QC envelope and the two regularization. Solutions of this PDE are strictly convex, and smoother than the robust-QC functions.