A numerical method for variational problems with convexity constraints

TL;DR

This paper introduces a fast numerical method for solving variational problems constrained by convexity, using polyhedral approximations to efficiently handle the convexity constraint.

Contribution

It proposes a novel approach that approximates the convex function cone with a polyhedral cone, enabling efficient linear optimization solutions.

Findings

Significantly faster computation compared to existing methods

Effective approximation of convexity constraints

Applicable to a range of variational problems

Abstract

We consider the problem of approximating the solution of variational problems subject to the constraint that the admissible functions must be convex. This problem is at the interface between convex analysis, convex optimization, variational problems, and partial differential equation techniques. The approach is to approximate the (non-polyhedral) cone of convex functions by a polyhedral cone which can be represented by linear inequalities. This approach leads to an optimization problem with linear constraints which can be computed efficiently, hundreds of times faster than existing methods.