On the algorithmic solution of optimization problems subject to probabilistic/robust (probust) constraints

• Published in: Mathematical methods of operations research (Heidelberg, Germany) Vol. 96; no. 1; pp. 1 - 37
• Main Authors: Berthold, Holger, Heitsch, Holger, Henrion, René, Schwientek, Jan
• Format: Journal Article
• Language: English
• Published: Berlin, Heidelberg Springer 01.08.2022; Springer Berlin Heidelberg; Springer Nature B.V
• Subjects: Adaptive algorithms; Adaptive discretization; Algorithms; Bilevel optimization; Business and Management; Calculus of Variations and Optimal Control; Optimization; Chance constraints; Grid refinement (mathematics); Mathematics; Mathematics and Statistics; Operations research; Operations Research/Decision Theory; Optimization; Original Article; Probabilistic constraints; Probability; Probust constraints; Reservoir management; Robustness (mathematics); Semi-infinite optimization; Reservoir management; Bilevel optimization; 65K05; Probabilistic constraints; Semi-infinite optimization; 90C17; Chance constraints; 90B05; 90C15; Probust constraints; Adaptive discretization
• ISSN: 1432-5217; 1432-2994; 1432-5217
• DOI: 10.1007/s00186-021-00764-8

We present an adaptive grid refinement algorithm to solve probabilistic optimization problems with infinitely many random constraints. Using a bilevel approach, we iteratively aggregate inequalities that provide most information not in a geometric but in a probabilistic sense. This conceptual idea, for which a convergence proof is provided, is then adapted to an implementable algorithm. The efficiency of our approach when compared to naive methods based on uniform grid refinement is illustrated for a numerical test example as well as for a water reservoir problem with joint probabilistic filling level constraints.