Efficient scalarization in multiobjective optimal control of a nonsmooth PDE

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2022
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This work deals with the efficient numerical characterization of Pareto stationary fronts for multiobjective optimal control problems with a moderate number of cost functionals and a mildly nonsmooth, elliptic, semilinear PDE-constraint. When “ample” controls are considered, strong stationarity conditions that can be used to numerically characterize the Pareto stationary fronts are known for our problem. We show that for finite dimensional controls, a sufficient adjoint-based stationarity system remains obtainable. It turns out that these stationarity conditions remain useful when numerically characterizing the fronts, because they correspond to strong stationarity systems for problems obtained by application of weighted-sum and reference point techniques to the multiobjective problem. We compare the performance of both scalarization techniques using quantifiable measures for the approximation quality. The subproblems of either method are solved with a line-search globalized pseudo-semismooth Newton method that appears to remove the degenerate behavior of the local version of the method employed previously. We apply a matrix-free, iterative approach to deal with the memory and complexity requirements when solving the subproblems of the reference point method and compare several preconditioning approaches.

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510 Mathematik
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Multiobjective optimal control, Nonsmooth optimization, Stationarity conditions, Pareto optimality, Scalarization methods, Pseudo-semismooth Newton method
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ISO 690BERNREUTHER, Marco, Georg MÜLLER, Stefan VOLKWEIN, 2022. Efficient scalarization in multiobjective optimal control of a nonsmooth PDE. In: Computational Optimization and Applications. Springer. 2022, 83(2), pp. 435-464. ISSN 0926-6003. eISSN 1573-2894. Available under: doi: 10.1007/s10589-022-00390-y
BibTex
@article{Bernreuther2022-11Effic-58473,
  year={2022},
  doi={10.1007/s10589-022-00390-y},
  title={Efficient scalarization in multiobjective optimal control of a nonsmooth PDE},
  number={2},
  volume={83},
  issn={0926-6003},
  journal={Computational Optimization and Applications},
  pages={435--464},
  author={Bernreuther, Marco and Müller, Georg and Volkwein, Stefan},
  note={German Research Foundation (DFG) under grant number VO 1658/5-2 within the priority program Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization (SPP 1962)}
}
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German Research Foundation (DFG) under grant number VO 1658/5-2 within the priority program Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization (SPP 1962)
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