On a Fully Adaptive SQP Method for PDAE-Constrained Optimal Control Problems with Control and State Constraints
We present an adaptive multilevel optimization approach which is suitable to solve complex real-world optimal control problems for time-dependent nonlinear partial differential algebraic equations with point-wise constraints on control and state. Relying on Moreau-Yosida regularization, the multilevel SQP method presented in Clever et al. (Generalized multilevel SQP-methods for PDAE-constrained optimization based on space-time adaptive PDAE solvers. In: Constrained optimization and optimal control for partial differential equations. Volume 160 of International series of numerical mathematics. Springer, Basel, pp 37–60, 2012) is extended to the state-constrained case. First-order convergence results are shown. The new multilevel SQP method is combined with the state-of-the-art software package KARDOS to allow the efficient resolution of different space and time scales in an adaptive manner. The numerical performance of the method is demonstrated and analyzed for a real-life three-dimensional radiative heat transfer problem modeling the cooling process in glass manufacturing and a two-dimensional thermistor problem modeling the heating process in steel hardening.
KeywordsPDAE-constrained optimization Multilevel optimization Generalized SQP method Trust region methods Control constraints State constraints Moreau-Yosida regularization Adaptive finite elements Rosenbrock methods Glass cooling Radiation Steel hardening
Mathematics Subject Classification (2010)49J20 49M25 65C20 65K10 65M60 90C46 90C55
The authors gratefully acknowledge the support of the German Research Foundation (DFG) within the Priority Program 1253 “Optimization with Partial Differential Equations” under grants LA1372/6-2 and UL158/7-2 and the support by the ‘Excellence Initiative’ of the German Federal and State Governments within the Graduate School of Computational Engineering at Technische Universität Darmstadt.
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