A BFGS-SQP method for nonsmooth, nonconvex, constrained optimization and its evaluation using relative minimization profiles

We propose an algorithm for solving nonsmooth, nonconvex, constrained optimization problems as well as a new set of visualization tools for comparing the performance of optimization algorithms. Our algorithm is a sequential quadratic optimization method that employs BFGS quasi-Newton Hessian approximations and an exact penalty function whose parameter is controlled using a steering strategy. We empirically validate our method using our new visualization tools, which we call relative minimization profiles. Such profiles are designed to simultaneously assess the relative performance of several algorithms with respect to three measures, highlighting the trade-offs between the measures when comparing algorithm performance on a heterogeneous test set. For example, in our experiments, we employ our algorithm to solve challenging test problems in controller design involving both locally Lipschitz and non-locally-Lipschitz functions, using relative minimization profiles to simultaneously compare our method against others in terms of objective quality, feasibility error, and speed of progress.