WIAS Preprint No. 3062, (2023)

Numerical solution of an optimal control problem with probabilistic and almost sure state constraints


  • Geiersbach, Caroline
    ORCID: 0000-0002-6518-7756
  • Henrion, René
    ORCID: 0000-0001-5572-7213
  • Pérez-Aroz, Pedro

2020 Mathematics Subject Classification

  • 49K20 49K45 35Q93 49J52 90C15


  • Stochastic optimization, risk averse PDE-constrained optimization under chance constraints, almost sure constraints, robust constraints




We consider the optimal control of a PDE with random source term subject to probabilistic or almost sure state constraints. In the main theoretical result, we provide an exact formula for the Clarke subdifferential of the probability function without a restrictive assumption made in an earlier paper. The focus of the paper is on numerical solution algorithms. As for probabilistic constraints, we apply the method of spherical radial decomposition. Almost sure constraints are dealt with a Moreau-Yosida smoothing of the constraint function accompanied by Monte Carlo sampling of the given distribution or its support or even just the boundary of its support. Moreover, one can understand the almost sure constraint as a probabilistic constraint with safety level one which offers yet another perspective. Finally, robust optimization can be applied efficiently when the support is sufficiently simple. A comparative study of these five different methodologies is carried out and illustrated.

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