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Journal of Industrial and Management Optimization (JIMO)
 

Line search globalization of a semismooth Newton method for operator equations in Hilbert spaces with applications in optimal control

Pages: 47 - 62, Volume 13, Issue 1, January 2017      doi:10.3934/jimo.2016003

 
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Matthias Gerdts - Institut für Mathematik und Rechneranwendung (LRT-1), Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577 Neubiberg/München, Germany (email)
Stefan Horn - Institut für Mathematik und Rechneranwendung (LRT-1), Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577 Neubiberg/München, Germany (email)
Sven-Joachim Kimmerle - Institut für Mathematik und Rechneranwendung (LRT-1), Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577 Neubiberg/München, Germany (email)

Abstract: We consider the numerical solution of nonlinear and nonsmooth operator equations in Hilbert spaces. A semismooth Newton method is used for search direction generation. The operator equation is solved by a globalized semismooth Newton method that is equipped with an Armijo linesearch using a semismooth merit function. We prove that an accumulation point of the globalized algorithm is a solution and transition to fast local convergence under a directional Hadamard-like continuity assumption on the Newton matrix. In particular, no auxiliary descent directions or smoothing steps are required. Finally, we apply this method to a control-constrained and also to a regularized state-constrained optimal control problem subject to partial differential equations.

Keywords:  Semismooth Newton method, line search globalization, superlinear convergence, optimal control of partial differential equations, control constraints, state constraints.
Mathematics Subject Classification:  Primary: 49J20, 49J52, 49M15; Secondary: 90C56, 65K10, 65N12.

Received: May 2014;      Revised: August 2015;      Available Online: March 2016.

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