Friedrich-Alexander-Universität Erlangen-Nürnberg
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We consider optimization problems with a joint probabilistic constraint under
normally distributed uncertain parameters. The parametric constraints
are replaced by one constraint stating that the probability of being feasible
shall exceed or be equal to a prescribed threshold. In order to apply the
concept to gas network optimization under uncertain boundary flows, which
corresponds to the demand of customers, we derive an analytic gradient formula.
The integral corresponding to the probability can be parameterized by
spherical radial decomposition. For this parameterization gradient formulas
are known under convexity assumptions of the parametric constraints in the
parameter. For the application in gas networks that we have in mind, the
convexity assumption of the parametric constraints is not satisfied. Therefore,
we weaken it to convexity of the region of feasible parameters for a fixed
optimization variable. We proceed to show that the assumptions needed for
the gradient formula are met in the gas network optimization problem on
a tree. For the numerical implementation we propose a multilevel sampling
algorithm that uses a coarse approximation of the chance constraint to generate
a warm start for the expensive approximation with fine sampling. The
numerical results illustrate that this approach significantly reduces the computation
time.