The semi-smooth Newton method for variationally discretized control constrained elliptic optimal control problems; implementation, convergence and globalization

Abstract

Combining the numerical concept of variational discretization and semi-smooth Newton methods for the numerical solution of pde-constrained optimization with control constraints, we place special emphasis on the implementation and globalization of the numerical algorithm. We prove fast local convergence of a globalized algorithm and illustrate our analytical and algorithmical findings by numerical experiments.

Keywords:

• variational discretization
• semi-smooth Newton method
• primal–dual active set strategy
• elliptic optimal control problem
• control constraints
• error estimates

Mathematics Subject Classification (2010) :

• 49J20
• 49K20
• 49M15

Acknowledgements

Michael Hinze gratefully acknowledges the support of the DFG Priority Program 1253 entitled Optimization With Partial Differential Equations. The authors wish to thank the anonymous referee for his valuable comments which helped them to improve the material of the paper, in particular, in Section 3.1. They also thank Andreas Günther for some fruitful discussions.