A difference of convex functions approach for sparse pde optimal control problems with nonconvex costs

TL;DR

This paper introduces a novel regularization approach for elliptic PDE optimal control problems with nonconvex costs, reformulating them as difference of convex functions problems to facilitate numerical solutions.

Contribution

It proposes a Huber type regularization that transforms nonconvex PDE control problems into DC programming problems, enabling the use of efficient DC algorithms.

Findings

Numerical experiments validate the effectiveness of the proposed regularization.

The method successfully approximates the original nonconvex control problems.

The approach provides necessary optimality conditions for the reformulated problems.

Abstract

We propose a local regularization of elliptic optimal control problems which involves the nonconvex $L^q$ fractional penalizations in the cost function. The proposed \emph{Huber type} regularization allows us to formulate the PDE constrained optimization formulation as a DC programming problem (difference of convex functions) that is useful to obtain necessary optimality conditions and tackle its numerical solution by applying the well known DC algorithm used in nonconvex optimization problems. By this procedure we approximate the original problem in terms of a consistent family of parameterized problems for which there are efficient numerical methods available. Finally, we present numerical experiments to illustrate our theory with different configurations associated to the parameters of the problem.