Okay, so now we have Elisa Strauch from the Technical University of Darmstadt and also

working on her PhD thesis and the title is Probabilistic Constraints in Optimization

Problems on Flow Networks.

Please, Elisa.

Yeah, thank you very much.

So I will talk about the probabilistic constraints which are used in optimization problems on

flow networks and this is a joint work with my supervisor Jens Lang and also with Martin

Gouart and Michael Schuster from the FAU Erlang Nürnberg.

So I will start with a short introduction to probabilistic constraints and I will present

two ways to compute the probability in these constraints.

The one way is the spherical radial decomposition and the other way is a direct approach which

we combined with a kernel density estimation.

And then we consider these two ways in the context of a stationary gas network and afterwards

I will extend the probabilistic constraints to a time-dependent setting and here we consider

a dynamic flow network.

And finally I will summarize the main points of my talk.

So let's start with the optimization problem.

There's a probabilistic constraint so we want to minimize objective function f and we have

now this probabilistic constraint here.

So we want that the probability of this inequality is larger or equal than alpha and alpha is

between 0 and 1 and here in this inequality we have now an n-dimensional random variable

Xi and now we want to compute this probability for a given x.

And now we consider at first the spherical radial decomposition.

Here we want to, here we consider at first a feasible set M and this is a set of all

Xi's for which this inequality is fulfilled.

So we can write the above probability in this term so we can also compute the probability

that Xi is in the set M and now the main point of the SRD is that we can write this probability

as an integral over the n-1-dimensional unit sphere.

And the other approach is a direct approach here we want to integrate the probability

density function of our random variable J so we have this x is given and in general

this density function is not known so we have to estimate this and for this estimation we

use a kernel density estimation.

So now let's start with the SRD method so we have a Xi that's our n-dimensional random

variable and here it's a Gaussian distributed random variable and then we can write this

probability that Xi is in the set M as an integral over the n-1-dimensional unit sphere

with a uniform distribution and we integrate the Xi distribution of this set.

So in the next slide I illustrate this set so here's our theorem and the integral over

the n-1-dimensional sphere and now we fix this W so W is an element of this sphere and

now we have to evaluate the Xi distribution along L times W up to the boundary of M.

So I try to illustrate this in this picture so we start at mu at the mean of the Gaussian

distribution and then we go along LW along the line up to the boundary of M and then

we evaluate the Xi distribution along this line and this value is this green area here.

And now the algorithm for the SRD is that okay we have our random variable and we have

this matrix L which we get out of a Cholesky decomposition of the covariance matrix sigma

and the first step is that we sample Q points uniformly distributed on the sphere and then

the second step is that for each sampling point we compute now a one-dimensional set

Mi so that was this line in this picture and then the third step is that we can now approximate

our desired probability by summing up over the sampling points and evaluating the Xi

distribution for this one-dimensional sets Mi.

So and now the direct approach here we want to integrate the probability density function