TY - JOUR A1 - Gonzalez Grandon, Tatiana A1 - Henrion, Rene A1 - Perez-Aros, Pedro T1 - Dynamic probabilistic constraints under continuous random distributions N2 - In this paper we address novel results on the theoretical structural analysis of dynamic joint probabilistic constraints under continuous random variables. This dynamic probabilistic function is important when decisions are time-dependent and when the modeler can react on past observations. We first study the continuity of dynamic probabilistic constraints and provide strong and weak semi-continuous results depending on whether the policies are supposed to be in the L^p or W^{1,p} spaces. Moreover, we prove the non-convexity of the feasible set of decisions induced by a dynamic probability function in the L^p space. Lastly, for a simple two-stage model, verifiable conditions for Lipschitz continuity and differentiability of this probability function are derived and endowed with explicit derivative formulae. Y1 - 2019 U6 - https://doi.org/10.1007/s10107-020-01593-z ER - TY - JOUR A1 - Hantoute, Abderrahim A1 - Henrion, Rene A1 - Perez-Aros, Pedro T1 - Subdifferential characterization of probability functions under Gaussian distribution N2 - Probability functions figure prominently in optimization problems of engineering. They may be nonsmooth even if all input data are smooth. This fact motivates the consideration of subdifferentials for such typically just continuous functions. The aim of this paper is to provide subdifferential formulae of such functions in the case of Gaussian distributions for possibly infinite-dimensional decision variables and nonsmooth (locally Lipschitzian) input data. These formulae are based on the spheric-radial decomposition of Gaussian random vectors on the one hand and on a cone of directions of moderate growth on the other. By successively adding additional hypotheses, conditions are satisfied under which the probability function is locally Lipschitzian or even differentiable. Y1 - 2018 U6 - https://doi.org/10.1007/s10107-018-1237-9 ER - TY - JOUR A1 - Geiersbach, Caroline A1 - Henrion, René A1 - Pérez-Aros, Pedro T1 - Numerical solution of an optimal control problem with probabilistic and almost sure state constraints N2 - We consider the optimal control of a PDE with random source term subject to probabilistic or almost sure state constraints. In the main theoretical result, we provide an exact formula for the Clarke subdifferential of the probability function without a restrictive assumption made in an earlier paper. The focus of the paper is on numerical solution algorithms. As for probabilistic constraints, we apply the method of spherical radial decomposition. Almost sure constraints are dealt with a Moreau--Yosida smoothing of the constraint function accompanied by Monte Carlo sampling of the given distribution or its support or even just the boundary of its support. Moreover, one can understand the almost sure constraint as a probabilistic constraint with safety level one which offers yet another perspective. Finally, robust optimization can be applied efficiently when the support is sufficiently simple. A comparative study of these five different methodologies is carried out and illustrated. Y1 - 2023 ER -