Robust Optimization for Unconstrained Simulation-Based Problems

In engineering design, an optimized solution often turns out to be suboptimal when errors are encountered. Although 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 that is suited for unconstrained problems with a nonconvex cost function as well as for problems based on simulations, such as large partial differential equations (PDE) solver, 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 nanophotonic 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.