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search.filters.applied.f.access: openAccess × Author: Henrion, René × Type: Text × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik × Subject: probabilistic constraints ×

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    Gradient formulae for nonlinear probabilistic constraints with Gaussian and aussian-like distributions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) van Ackooij, Wim; Henrion, René
    Probabilistic constraints represent a major model of stochastic optimization. A possible approach for solving probabilistically constrained optimization problems consists in applying nonlinear programming methods. In order to do so, one has to provide sufficiently precise approximations for values and gradients of probability functions. For linear probabilistic constraints under Gaussian distribution this can be successfully done by analytically reducing these values and gradients to values of Gaussian distribution functions and computing the latter, for instance, by Genz’ code. For nonlinear models one may fall back on the spherical-radial decomposition of Gaussian random vectors and apply, for instance, Deák’s sampling scheme for the uniform distribution on the sphere in order to compute values of corresponding probability functions. The present paper demonstrates how the same sampling scheme can be used in order to simultaneously compute gradients of these probability functions. More precisely, we prove a formula representing these gradients in the Gaussian case as a certain integral over the sphere again. Later, the result is extended to alternative distributions with an emphasis on the multivariate Student (or T-) distribution.
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    Generalized gradients for probabilistic/robust (probust) constraints
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Ackooij, Wim van; Henrion, René; Pérez-Aros, Pedro
    Probability 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.
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    Optimal Neumann boundary control of a vibrating string with uncertain initial data and probabilistic terminal constraints
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Farshbaf Shaker, Mohammad Hassan; Gugat, Martin; Heitsch, Holger; Henrion, René
    In optimal control problems, often initial data are required that are not known exactly in practice. In order to take into account this uncertainty, we consider optimal control problems for a system with an uncertain initial state. A finite terminal time is given. On account of the uncertainty of the initial state, it is not possible to prescribe an exact terminal state. Instead, we are looking for controls that steer the system into a given neighborhood of the desired terminal state with sufficiently high probability. This neighborhood is described in terms of an inequality for the terminal energy. The probabilistic constraint in the considered optimal control problem leads to optimal controls that are robust against the inevitable uncertainties of the initial state. We show the existence of such optimal controls. Numerical examples with optimal Neumann control of the wave equation are presented.
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    Joint model of probabilistic-robust (probust) constraints with application to gas network optimization
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Adelhütte, Dennis; Aßmann, Denis; Grandón, Tatiana González; Gugat, Martin; Heitsch, Holger; Henrion, René; Liers, Frauke; Nitsche, Sabrina; Schultz, Rüdiger; Stingl, Michael; Wintergerst, David
    Optimization problems under uncertain conditions abound in many real-life applications. While solution approaches for probabilistic constraints are often developed in case the uncertainties can be assumed to follow a certain probability distribution, robust approaches are usually applied in case solutions are sought that are feasible for all realizations of uncertainties within some predefined uncertainty set. As many applications contain different types of uncertainties that require robust as well as probabilistic treatments, we introduce a class of joint probabilistic/robust constraints. Focusing on complex uncertain gas network optimization problems, we show the relevance of this class of problems for the task of maximizing free booked capacities in an algebraic model for a stationary gas network. We furthermore present approaches for finding their solution. Finally, we study the problem of controlling a transient system that is governed by the wave equation. The task consists in determining controls such that a certain robustness measure remains below some given upper bound with high probability.