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A BFGS-SQP Method for Nonsmooth, Nonconvex, Constrained Optimization and its Evaluation using Relative Minimization Profiles

Frank E. Curtis (frank.e.curtis***at***gmail.com)
Tim Mitchell (tim.mitchell***at***cims.nyu.edu)
Michael L. Overton (overton***at***cims.nyu.edu)

Abstract: 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 Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton Hessian approximations and an exact penalty function whose parameter is controlled using a steering strategy. While our method has no convergence guarantees, we have found it to perform very well in practice on challenging test problems in controller design involving both locally Lipschitz and non-locally-Lipschitz objective and constraint functions with constraints that are typically active at local minimizers. In order to empirically validate and compare our method with available alternativesóon a new test set of 200 problems of varying sizesówe employ 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 objective quality, feasibility, and speed of progress, highlighting the trade-offs between these measures when comparing algorithm performance.

Keywords: nonconvex optimization; nonsmooth optimization; constrained optimization; sequential quadratic programming; exact penalty methods; benchmarking; performance profiles; computational budget

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Category 2: Optimization Software and Modeling Systems (Optimization Software Benchmark )

Category 3: Applications -- Science and Engineering (Control Applications )

Citation: F. E. Curtis, T. Mitchell, and M. L. Overton. A BFGS-SQP Method for Nonsmooth, Nonconvex, Constrained Optimization and its Evaluation using Relative Minimization Profiles. Optimization Methods and Software, 32(1):148-181, 2017.

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Entry Submitted: 06/23/2015
Entry Accepted: 06/23/2015
Entry Last Modified: 01/02/2017

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