There are many algorithms for minimization when the objective function is differentiable, convex, or has some other known structure, but few options when none of the above hold, particularly when the objective function is nonsmooth at minimizers, as is often the case in applications. We discuss two algorithms for minimization of nonsmooth, nonconvex functions. Gradient Sampling is a simple method that, although computationally intensive, has a nice convergence theory. The method is robust and the convergence theory has recently been extended to constrained problems. BFGS is a well known method, developed for smooth problems, but which is remarkably effective for nonsmooth problems too. Although our theoretical results in the nonsmooth case are quite limited, we have made some remarkable empirical observations and have had broad success in applications. Limited Memory BFGS is a popular extension for large problems, and it is also applicable to the nonsmooth case, although our experience with it is more mixed.
Michael L. Overton is Professor of Computer Science and Mathematics and chair of the Computer Science Department at the Courant Institute of Mathematical Sciences, New York University. He received his Ph.D. in Computer Science from Stanford University in 1979. He is a fellow of SIAM (Society for Industrial and Applied Mathematics) and of the IMA (Institute of Mathematics and its Applications, UK). He served on the Council and Board of Trustees of SIAM from 1991 to 2005, including a term as Chair of the Board from 2004 to 2005. He is a member of the Council of FoCM (Foundations of Computational Mathematics), served on the Board of Directors of the Canadian Mathematical Society from 2001 to 2005, on the Scientific Advisory Board of the Fields Institute from 2001 to 2004, and is currently on the Scientific Review Committee for the Pacific Institute for the Mathematical Sciences. He served as Editor-in-Chief of SIAM Journal on Optimization from 1995 to 1999 and of the IMA Journal of Numerical Analysis from 2007 to 2008, and was the Editor-in-Chief of the MPS(Mathematical Programming Society)-SIAM joint book series from 2003 to 2007. He currently serves on the editorial boards of SIAM Journal on Matrix Analysis and Applications, Numerische Mathematik and the IMA Journal of Numerical Analysis, as well as the editorial advisory board for Foundations of Computational Mathematics. His research interests are at the interface of optimization and linear algebra, especially nonsmooth optimization problems involving eigenvalues, pseudospectra, stability and robust control. He is the author of “Numerical Computing with IEEE Floating Point Arithmetic” (SIAM, 2001).