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- ItemGeneralized gradients for probabilistic/robust (probust) constraints(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Ackooij, Wim van; Henrion, René; Pérez-Aros, PedroProbability functions are a powerful modelling tool when seeking to account for uncertainty in optimization problems. In practice, such uncertainty may result from different sources for which unequal information is available. A convenient combination with ideas from robust optimization then leads to probust functions, i.e., probability functions acting on generalized semi-infinite inequality systems. In this paper we employ the powerful variational tools developed by Boris Mordukhovich to study generalized differentiation of such probust functions. We also provide explicit outer estimates of the generalized subdifferentials in terms of nominal data.
- ItemOn the algorithmic solution of optimization problems subject to probabilistic/robust (probust) constraints(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Berthold, Holger; Heitsch, Holger; Henrion, René; Schwientek, JanWe 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.