Probabilistic Constrained Optimization on Flow Networks

Michael Schuster, FAU, Department Mathematik

Abstract: Uncertainty often plays an important role in dynamic flow problems. In this paper, we consider both, a stationary and a dynamic flow model with

uncertain boundary data on networks. We introduce two different ways how to compute the probability for random boundary data to be feasible, discussing their advantages and disadvantages. In this context, feasible means, that the flow corresponding to the random boundary data meets some box constraints at the network junctions. The first method is the spheric radial decomposition and the second method is a kernel density estimator to compute the density of the solution at the nodes.

In both settings, we consider certain optimization problems and we compute the derivative of the probabilistic constraint using the kernel density estimator. Moreover, we derive necessary optimality conditions for the stationary and the

dynamic case. Throughout the paper, we use numerical examples to illustrate our results by comparing them with a classical Monte Carlo approach to ompute the desired probability.

Key-words: Stochastic Optimization, Probabilistic Constraints, Uncertain Boundary Data, Spheric-Radial Decomposition, Kerne-Density Estimator, Flow Networks, Gas Networks, Contamination of Water