Gradient estimates for Gaussian distribution functions: application to probabilistically constrained optimization problems
René Henrion
Numerical Algebra, Control & Optimization 2012, 2(4): 655-668 doi: 10.3934/naco.2012.2.655
We provide lower estimates for the norm of gradients of Gaussian distribution functions and apply the results obtained to a special class of probabilistically constrained optimization problems. In particular, it is shown how the precision of computing gradients in such problems can be controlled by the precision of function values for Gaussian distribution functions. Moreover, a sensitivity result for optimal values with respect to perturbations of the underlying random vector is derived. It is shown that the so-called maximal increasing slope of the optimal value with respect to the Kolmogorov distance between original and perturbed distribution can be estimated explicitly from the input data of the problem.
keywords: Gaussian distribution function sensitivity of optimal values. stochastic optimization Probabilistic constraints chance constraints
Path planning and collision avoidance for robots
Matthias Gerdts René Henrion Dietmar Hömberg Chantal Landry
Numerical Algebra, Control & Optimization 2012, 2(3): 437-463 doi: 10.3934/naco.2012.2.437
An optimal control problem to find the fastest collision-free trajectory of a robot surrounded by obstacles is presented. The collision avoidance is based on linear programming arguments and expressed as state constraints. The optimal control problem is solved with a sequential programming method. In order to decrease the number of unknowns and constraints a backface culling active set strategy is added to the resolution technique.
keywords: collision avoidance cooperative robots backface culling active set strategy. Optimal control
Discrepancy distances and scenario reduction in two-stage stochastic mixed-integer programming
René Henrion Christian Küchler Werner Römisch
Journal of Industrial & Management Optimization 2008, 4(2): 363-384 doi: 10.3934/jimo.2008.4.363
Polyhedral discrepancies are relevant for the quantitative stability of mixed-integer two-stage and chance constrained stochastic programs. We study the problem of optimal scenario reduction for a discrete probability distribution with respect to certain polyhedral discrepancies and develop algorithms for determining the optimally reduced distribution approximately. Encouraging numerical experience for optimal scenario reduction is provided.
keywords: Kolmogorov metric. chance constraints scenario reduction discrepancy two-stage Stochastic programming mixed-integer

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