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

• Published in: Journal of optimization theory and applications Vol. 204; no. 1; p. 7
• Main Authors: Geiersbach, Caroline, Henrion, René, Pérez-Aros, Pedro
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.01.2025; Springer Nature B.V
• Subjects: 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
• DOI: 10.1007/s10957-024-02578-0

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.