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Version: publishedVersion × Subject: semi-infinite optimization ×

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    On 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, Jan
    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.
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    Stability and sensitivity of optimization problems with first order stochastic dominance constraints
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Dentcheva, Darinka; Henrion, René; Ruszczynski, Andrzej
    We analyze the stability and sensitivity of stochastic optimization problems with stochastic dominance constraints of first order. We consider general perturbations of the underlying probability measures in the space of regular measures equipped with a suitable discrepancy distance. We show that the graph of the feasible set mapping is closed under rather general assumptions. We obtain conditions for the continuity of the optimal value and upper-semicontinuity of the optimal solutions, as well as quantitative stability estimates of Lipschitz type. Furthermore, we analyze the sensitivity of the optimal value and obtain upper and lower bounds for the directional derivatives of the optimal value. The estimates are formulated in terms of the dual utility functions associated with the dominance constraints.