On convex functions and the finite element method

TL;DR

This paper introduces a finite element approach to approximate convex functions, addressing challenges of non-smooth solutions and Hessian discretization, with proven convergence and practical examples using semidefinite programming.

Contribution

It proposes a novel finite element discretization of the Hessian for convex functions, ensuring convergence even for non-smooth solutions with minimal constraints.

Findings

Convergence proven under general conditions

Effective finite element discretization of the Hessian

Practical examples using semidefinite programming

Abstract

Many problems of theoretical and practical interest involve finding a convex or concave function. For instance, optimization problems such as finding the projection on the convex functions in $H^k(\Omega)$, or some problems in economics. In the continuous setting and assuming smoothness, the convexity constraints may be given locally by asking the Hessian matrix to be positive semidefinite, but in making discrete approximations two difficulties arise: the continuous solutions may be not smooth, and an adequate discrete version of the Hessian must be given. In this paper we propose a finite element description of the Hessian, and prove convergence under very general conditions, even when the continuous solution is not smooth, working on any dimension, and requiring a linear number of constraints in the number of nodes. Using semidefinite programming codes, we show concrete examples of approximations to optimization problems.