In engineering design, an optimized solution often turns out to be suboptimal, when errors are encountered. While the theory of robust convex optimization has taken significant strides over the past decade, all approaches fail if the underlying cost function is not explicitly given; it is even worse if the cost function is nonconvex. In this work, we present a robust optimization method, which is suited for unconstrained problems with a nonconvex cost function as well as for problems based on simulations such as large PDE solvers, response surface, and kriging metamodels. Moreover, this technique can be employed for most real-world problems, because it operates directly on the response surface and does not assume any specific structure of the problem. We present this algorithm along with the application to an actual engineering problem in electromagnetic multiple-scattering of aperiodically arranged dielectrics, relevant to nano-photonic design. The corresponding objective function is highly nonconvex and resides in a 100-dimensional design space. Starting from an "optimized" design, we report a robust solution with a significantly lower worst case cost, while maintaining optimality. We further generalize this algorithm to address a nonconvex optimization problem under both implementation errors and parameter uncertainties.

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W e propose a new robust optimization method for problems with objective functions that may be computed via numerical simulations and incorporate constraints that need to be feasible under perturbations. The proposed method iteratively moves along descent directions for the robust problem with nonconvex constraints and terminates at a robust local minimum. We generalize the algorithm further to model parameter uncertainties. We demonstrate the practicability of the method in a test application on a nonconvex problem with a polynomial cost function as well as in a real-world application to the optimization problem of intensity-modulated radiation therapy for cancer treatment. The method significantly improves the robustness for both designs.

W ith growing dependence of many cities on urban mass transit, even limited disruptions of public transportation networks can lead to widespread confusion and significant productivity losses. A need exists for systematic approaches to developing efficient responses to minimize such negative impacts. We present an optimization-based approach that responds to degradations of urban transit rail networks by introducing smartly designed bus bridging services that take into consideration commuter travel demand at the time of the disruption. The approach consists of three fundamental steps, namely, (1) a column generation procedure to dynamically generate demand-responsive candidate bus routes, (2) a path-based multicommodity network flow model to identify the most effective combination of these candidate bus routes, and (3) another optimizationbased procedure to determine simultaneously the optimal allocation of available vehicle resources among the selected routes and corresponding headways. The approach is applied to two case studies defined using actual data. The results show that the proposed approach can be carried out efficiently and that adding nonintuitive bus routes to the standard bus bridging services can significantly reduce the average travel delay. Moreover, the approach distributes delay more equitably. Many realistic operating constraints can also be handled.