Coupled versus decoupled penalization of control complementarity constraints

¹ Faculty of Mathematics and Computer Science, Technische Universität Bergakademie Freiberg, Germany.
² Institute of Mathematics, Chair of Optimal Control, Brandenburgische Technische Universität Cottbus-Senftenberg, Germany.

^* Corresponding author: yu.deng@math.tu-freiberg.de

Received: 28 October 2019
Accepted: 21 February 2021

Abstract

This paper deals with the numerical solution of optimal control problems with control complementarity constraints. For that purpose, we suggest the use of several penalty methods which differ with respect to the handling of the complementarity constraint which is either penalized as a whole with the aid of NCP-functions or decoupled in such a way that non-negativity constraints as well as the equilibrium condition are penalized individually. We first present general global and local convergence results which cover several different penalty schemes before two decoupled methods which are based on a classical ℓ₁- and ℓ₂-penalty term, respectively, are investigated in more detail. Afterwards, the numerical implementation of these penalty methods is discussed. Based on some examples, where the optimal boundary control of a parabolic partial differential equation is considered, some quantitative properties of the resulting algorithms are compared.