A partial differential equation for the strictly quasiconvex envelope
For researchers in PDEs and optimization, this provides a numerically tractable regularization for quasiconvex envelope problems, though it is an incremental improvement over existing work.
The authors introduce a stronger regularization for the quasiconvex envelope PDE, enabling convergent finite difference approximations. The resulting solutions are strictly convex and smoother than prior robust-QC functions.
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.