Kuchlbauer M, Liers F, Stingl M (2022)
Publication Status: Accepted
Publication Type: Journal article
Future Publication Type: Journal article
Publication year: 2022
Pages Range: 1056–1086
Journal Issue: 195
URI: https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/414
DOI: 10.1007/s10957-022-02114-y
Open Access Link: https://doi.org/10.1007/s10957-022-02114-y
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.
APA:
Kuchlbauer, M., Liers, F., & Stingl, M. (2022). Outer approximation for mixed-integer nonlinear robust optimization. Journal of Optimization Theory and Applications, 195, 1056–1086. https://dx.doi.org/10.1007/s10957-022-02114-y
MLA:
Kuchlbauer, Martina, Frauke Liers, and Michael Stingl. "Outer approximation for mixed-integer nonlinear robust optimization." Journal of Optimization Theory and Applications 195 (2022): 1056–1086.
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