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Author: Heitsch, Holger × Start date: 2010 × DDC: 510 × Subject: optimality conditions × WGL Institute: WIAS ×

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    Solving joint chance constrained problems using regularization and Benders decomposition
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Adam, Lukás; Branda, Martin; Heitsch, Holger; Henrion, René
    In this paper we investigate stochastic programms with joint chance constraints. We consider discrete scenario set and reformulate the problem by adding auxiliary variables. Since the resulting problem has a difficult feasible set, we regularize it. To decrease the dependence on the scenario number, we propose a numerical method by iteratively solving a master problem while adding Benders cuts. We find the solution of the slave problem (generating the Benders cuts) in a closed form and propose a heuristic method to decrease the number of cuts. We perform a numerical study by increasing the number of scenarios and compare our solution with a solution obtained by solving the same problem with continuous distribution.
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    An enumerative formula for the spherical cap discrepancy
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Heitsch, Holger; Henrion, René
    The spherical cap discrepancy is a widely used measure for how uniformly a sample of points on the sphere is distributed. Being hard to compute, this discrepancy measure is typically replaced by some lower or upper estimates when designing optimal sampling schemes for the uniform distribution on the sphere. In this paper, we provide a fully explicit, easy to implement enumerative formula for the spherical cap discrepancy. Not surprisingly, this formula is of combinatorial nature and, thus, its application is limited to spheres of small dimension and moderate sample sizes. Nonetheless, it may serve as a useful calibrating tool for testing the efficiency of sampling schemes and its explicit character might be useful also to establish necessary optimality conditions when minimizing the discrepancy with respect to a sample of given size.