A semismooth Newton method for a class of semilinear optimal control problems with box and volume constraints

Authors:
Samuel Amstutz;Antoine Laurain
Affiliations:
Laboratoire de Mathématiques d'Avignon, Faculté des Sciences, Université d'Avignon, Avignon, France 84000;Department of Mathematics, Technical University of Berlin, Berlin, Germany
Venue:
Computational Optimization and Applications
Year:
2013

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Abstract

In this paper we consider optimal control problems subject to a semilinear elliptic state equation together with the control constraints 0≤u≤1 and 驴u=m. Optimality conditions for this problem are derived and reformulated as a nonlinear, nonsmooth equation which is solved using a semismooth Newton method. A regularization of the nonsmooth equation is necessary to obtain the superlinear convergence of the semismooth Newton method. We prove that the solutions of the regularized problems converge to a solution of the original problem and a path-following technique is used to ensure a constant decrease rate of the residual. We show that, in certain situations, the optimal controls take 0---1 values, which amounts to solving a topology optimization problem with volume constraint.