Numerical Solution of an Optimal Control Problem with Probabilistic and Almost Sure State 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...

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Vydáno v:Journal of optimization theory and applications Ročník 204; číslo 1; s. 7
Hlavní autoři: Geiersbach, Caroline, Henrion, René, Pérez-Aros, Pedro
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.01.2025
Springer Nature B.V
Témata:
Algorithms
Applications of Mathematics
Approximation
Calculus of Variations and Optimal Control; Optimization
Comparative studies
Constraints
Decomposition
Engineering
Investigations
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimal control
Optimization
Probability theory
Risk aversion
Statistical analysis
Theory of Computation
49K45
Risk averse PDE-constrained optimization under uncertainty
49K20
49J52
Robust constraints
Chance constraints
90C15
Stochastic optimization
Almost sure constraints
35Q93
ISSN:0022-3239, 1573-2878
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Popis
Shrnutí: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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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-024-02578-0