WIAS Preprint No. 3021, (2023)

Optimality conditions in control problems with random state constraints in probabilistic or almost-sure form


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

2020 Mathematics Subject Classification

  • 49K20 49K45 35Q93 49J52 90C15


  • Optimality conditions, stochastic optimization, PDE-constrained optimization under uncertainty, chance constraints, almost sure constraints, robust constraints




In this paper, we discuss optimality conditions for optimization problems subject to random state constraints, which are modeled in probabilistic or almost sure form. While the latter can be understood as the limiting case of the former, the derivation of optimality conditions requires substantially different approaches. We apply them to a linear elliptic partial differential equation (PDE) with random inputs. In the probabilistic case, we rely on the spherical-radial decomposition of Gaussian random vectors in order to formulate fully explicit optimality conditions involving a spherical integral. In the almost sure case, we derive optimality conditions and compare them to a model based on robust constraints with respect to the (compact) support of the given distribution.

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