Kuchlbauer M, Liers F, Stingl M (2022)
Publication Language: English
Publication Type: Journal article
Publication year: 2022
Book Volume: 34
Pages Range: 2106 - 2124
Journal Issue: 4
Currently, there are few theoretical or practical approaches available for general nonlinear robust optimization. Moreover, the approaches that do exist impose restrictive assumptions on the problem structure. We present an adaptive bundle method for nonlinear and non-convex robust optimization problems with a suitable notion of inexactness in function values and subgradients. As the worst case evaluation requires a global solution to the adversarial problem, it is a main challenge in a general non-convex nonlinear setting. Moreover, computing elements of an epsilon-perturbation of the Clarke subdifferential in the l2-norm sense is in general prohibitive for this class of problems. In this article, instead of developing an entirely new bundle concept, we demonstrate how existing approaches, such as Noll's bundle method for non-convex minimization with inexact information (Computational and analytical mathematics 50: 555-592, 2013) can be modified to be able to cope with this situation. Extending the non-convex bundle concept to the case of robust optimization in this way, we prove convergence under two assumptions: Firstly, that the objective function is lower C1 and secondly, that approximately optimal solutions to the adversarial maximization problem are available. The proposed method is hence applicable to a rather general setting of nonlinear robust optimization problems. In particular, we do not rely on a specific structure of the adversary's constraints. The considered class of robust optimization problems covers the case that the worst-case adversary only needs to be evaluated up to a certain precision. One possibility to evaluate the worst case with the desired degree of precision is the use of techniques from mixed-integer linear programming (MIP).
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 method.
Kuchlbauer, M., Liers, F., & Stingl, M. (2022). Adaptive bundle methods for nonlinear robust optimization. Informs Journal on Computing, 34(4), 2106 - 2124. https://dx.doi.org/10.1287/ijoc.2021.1122
Kuchlbauer, Martina, Frauke Liers, and Michael Stingl. "Adaptive bundle methods for nonlinear robust optimization." Informs Journal on Computing 34.4 (2022): 2106 - 2124.