Adaptive Bundle Methods for Nonlinear Robust Optimization

Abstract

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 nonconvex 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 nonconvex nonlinear setting. Moreover, computing elements of an ε-perturbation of the Clarke subdifferential in the [Formula: see text]-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 nonconvex minimization with inexact information [Noll D (2013) Bundle method for non-convex minimization with inexact subgradients and function values. Computational and Analytical Mathematics, Springer Proceedings Mathematics, vol. 50 (Springer, New York), 555–592.] can be modified to be able to cope with this situation. Extending the nonconvex bundle concept to the case of robust optimization in this way, we prove convergence under two assumptions: first, that the objective function is lower C1 and, second, 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. 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. Summary of Contribution: Nonlinear robust optimization is a relevant field of research as real-world optimization problems usually suffer from not precisely known parameters, for example, physical parameters that cannot be measured exactly. Currently, there are few theoretical or practical approaches available for general nonlinear robust optimization. Moreover, the methods that do exist impose restrictive assumptions on the problem structure. Writing nonlinear robust optimization tasks in minimax form, in principle, bundle methods can be used to solve the resulting nonsmooth problem. However, there are a number of difficulties to overcome. First, the inner adversarial problem needs to be solved to global optimality, which is a major challenge in a general nonconvex nonlinear setting. In order to cope with this, an adaptive solution approach, which allows for inexactness, is required. A second challenge is then that the computation of elements from an ε-neighborhood of the Clarke subdifferential is, in general, prohibitive. We show how an existing bundle concept by D. Noll for nonconvex problems with inexactness in function values and subgradients can be adapted to this situation. The resulting method only requires availability of approximate worst-case evaluations, and in particular, it does not rely on a specific structure of the adversarial constraints. To evaluate the worst case with the desired degree of precision, one possibility is the use of techniques from mixed-integer linear programming. In the course of the paper, we discuss convergence properties of the resulting method and demonstrate its efficiency by means of robust gas transport problems.