Proximal Methods for Elliptic Optimal Control Problems with Sparsity Cost Functional

Andreas Schindele,Alfio Borzì

doi:10.4236/am.2016.79086

Abstract

First-order proximal methods that solve linear and bilinear elliptic optimal control problems with a sparsity cost functional are discussed. In particular, fast convergence of these methods is proved. For benchmarking purposes, inexact proximal schemes are compared to an inexact semismooth Newton method. Results of numerical experiments are presented to demonstrate the computational effectiveness of proximal schemes applied to infinite-dimensional elliptic optimal control problems and to validate the theoretical estimates.

Highlights

IntroductionA great research effort has been made to solve optimization problems governed by Partial
In recent years, a great research effort has been made to solve optimization problems governed by PartialDifferential Equations (PDEs); see, e.g., [1]-[3] and references therein
More recently, the investigation of L1 cost functionals has become a central topic in PartialDifferential Equations (PDEs)-based optimization [4]-[6], because they give rise to sparse controls that are advantageous in many applications like optimal actuator placement [4] or impulse control [7]

Summary

Introduction

A great research effort has been made to solve optimization problems governed by Partial. Differential Equations (PDEs); see, e.g., [1]-[3] and references therein. This research has focused on objective functionals with differentiable L2 terms and non-smoothness resulted from the presence of control and state constraints. More recently, the investigation of L1 cost functionals has become a central topic in PDE-based optimization [4]-[6], because they give rise to sparse controls that are advantageous in many applications like optimal actuator placement [4] or impulse control [7]. A representative formulation of optimal control problems with L1 control costs is the following min ( y,u )∈H (Ω)× L2 (Ω) s.t.

Objectives

Methods

Conclusion