## Section: New Results

### Robust/Stochastic programming

**Locating stations in a one-way electric car sharing system under demand uncertainty:**
In [16], we focused on a problem of locating recharging stations in one-way station based electric car sharing systems which operate under demand uncertainty. We modeled this problem as a mixed integer stochastic program and develop a Benders decomposition algorithm based on this formulation. We integrated a stabilization procedure to our algorithm and conduct a large-scale experimental study on our methods.
To conduct the computational experiments, we developed a demand forecasting method allowing to generate many demand scenarios.
The method was applied to real data from Manhattan taxi trips.

**Bookings in the European Gas Market: Characterisation of Feasibility and Computational Complexity Results:**
As a consequence of the liberalisation of the European gas market in the last decades, gas trading and transport have been decoupled. At the core of this decoupling are so-called bookings and nominations. Bookings are special long-term capacity right contracts that guarantee that a specified amount of gas can be supplied or withdrawn at certain entry or exit nodes of the network. These supplies and withdrawals are nominated at the day-ahead. These bookings then need to be feasible, i.e., every nomination that complies with the given bookings can be transported. While checking the feasibility of a nomination can typically be done by solving a mixed-integer nonlinear feasibility problem, the verification of feasibility of a set of bookings is much harder. We consider the question of how to verify the feasibility of given bookings for a number of special cases. For our physics model we impose a steady-state potential-based flow model and disregard controllable network elements. We derive a characterisation of feasible bookings, which is then used to show that the problem is in coNP for the general case but can be solved in polynomial time for linear potential-based flow models. Moreover, we present a dynamic programming approach for deciding the feasibility of a booking in tree-shaped networks even for nonlinear flow models [25]. Further, in [71], we show that the feasibility of a booking also can be decided in polynomial time on single-cycle networks.

**Robust bilevel programs:**
Bilevel optimization problems embed the optimality conditions of a sub-problem into the constraints of
a decision-making process. A general question of bilevel optimization occurs where the lower-level is
solved (only) to near-optimality. Solving bilevel problems under limited deviations of the lower-level
variables was introduced under the term “$\u03f5$-approximation” of the pessimistic bilevel problem.
In [77]
the authors define special properties and a solution method for this variant in the so-called independent
case, i.e., where the lower-level feasible set is independent of the upper-level decision. In [66], we generalized the approach of *Wiesemann et al. 2013*, to problems with constraints involving upper- and lower-level
variables in the constraints at both levels. The purpose of this generalization is to protect the upper-level
feasibility against uncertainty of near-optimal solutions of the lower-level. We call this near-optimal
robustness and the generalization is a near-optimal robust bilevel problem (NORBiP). NORBiP is a
bilinear bilevel problem, and this makes it very hard in general. We have defined and implemented a solution algorithm for the linear linear NORBiP [66].