We investigate the procedure on some analytic examples. As applications, we study the gas transport problem under uncertainties in demand and in physical parameters that affect pressure losses in the pipes. Computational results for examples in large realistic gas network instances demonstrate the applicability as well as the efficiency of the metho}, author = {Kuchlbauer, Martina and Liers, Frauke and Stingl, Michael}, doi = {10.1287/ijoc.2021.1122}, faupublication = {yes}, journal = {Informs Journal on Computing}, keywords = {bundle method; nonlinear robust optimization; inexactness; minimax; gas transport problem}, pages = {2106 - 2124}, peerreviewed = {Yes}, title = {{Adaptive} bundle methods for nonlinear robust optimization}, volume = {34}, year = {2022} } @article{faucris.262995087, abstract = {Currently, few approaches are available for general nonlinear robust optimization. Those that do exist typically require restrictive assumptions on the adversarial problem or do not guarantee robust protection. To address this, we present an algorithm that combines outer approximation with a bundle method. This algorithm is applicable to convex mixed-integer nonlinear robust optimization problems and necessitates only inexact worst-case evaluations. A key feature of this method is that it does not rely on a specific structure of the adversarial problem and allows it to be non-convex. A major challenge of such a general nonlinear setting is ensuring robust protection, as this calls for a global solution of the adversarial problem. However, our method is able to achieve this, requiring worst-case evaluations only up to a certain precision. For example, the necessary assumptions can be met by approximating a non-convex adversarial problem via piecewise linearization and solving the resulting problem up to any requested error as a mixed-integer linear problem.

We model a robust optimization problem as a nonsmooth mixed-integer nonlinear problem and tackle it adopting an outer approximation approach that requires only inexact function values and subgradients. To deal with the arising nonlinear subproblems, we render an adaptive bundle method applicable to this setting. Relying on its convergence to approximate critical points, we prove, as a consequence, finite convergence of the outer approximation approach.

As an application, we study the gas transport problem under uncertainties on realistic instances and provide computational results demonstrating the efficiency of our method}, author = {Kuchlbauer, Martina and Liers, Frauke and Stingl, Michael}, doi = {10.1007/s10957-022-02114-y}, faupublication = {yes}, journal = {Journal of Optimization Theory and Applications}, keywords = {robust optimization; mixed-integer nonlinear optimization; outer approximation; bundle method; gas transport problem}, pages = {1056–1086}, peerreviewed = {Yes}, title = {{Outer} approximation for mixed-integer nonlinear robust optimization}, url = {https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/414}, year = {2022} } @article{faucris.283625211, abstract = {Pareto efficiency for robust linear programs was introduced by Iancu and Trichakis in [Manage Sci 60(1):130-147, 9]. We generalize their approach and theoretical results to robust optimization problems in Euclidean spaces with affine uncertainty. Additionally, we demonstrate the value of this approach in an exemplary manner in the area of robust semidefinite programming (SDP). In particular, we prove that computing a Pareto robustly optimal solution for a robust SDP is tractable and illustrate the benefit of such solutions at the example of the maximal eigenvalue problem. Furthermore, we modify the famous algorithm of Goemans and Williamson [Assoc Comput Mach 42(6):1115-1145, 8] in order to compute cuts for the robust max-cut problem that yield an improved approximation guarantee in non-worst-case scenarios.}, author = {Adelhütte, Dennis and Biefel, Christian and Kuchlbauer, Martina and Rolfes, Jan}, doi = {10.1007/s11590-022-01929-y}, faupublication = {yes}, journal = {Optimization Letters}, note = {CRIS-Team WoS Importer:2022-10-21}, peerreviewed = {Yes}, title = {{Pareto} robust optimization on {Euclidean} vector spaces}, year = {2022} } @article{faucris.263647932, abstract = {Pareto efficiency for robust linear programs was introduced by Iancu and Trichakis in [Manage Sci 60(1):130–147, 9]. We generalize their approach and theoretical results to robust optimization problems in Euclidean spaces with affine uncertainty. Additionally, we demonstrate the value of this approach in an exemplary manner in the area of robust semidefinite programming (SDP). In particular, we prove that computing a Pareto robustly optimal solution for a robust SDP is tractable and illustrate the benefit of such solutions at the example of the maximal eigenvalue problem. Furthermore, we modify the famous algorithm of Goemans and Williamson [Assoc Comput Mach 42(6):1115–1145, 8] in order to compute cuts for the robust max-cut problem that yield an improved approximation guarantee in non-worst-case scenarios.