Behavior of Limited Memory BFGS when Applied to Nonsmooth Functions and their Nesterov Smoothings.

The motivation to study the behavior of limited-memory BFGS (L-BFGS) on nonsmooth optimization problems is based on two empirical observations: the widespread success of L-BFGS in solving large-scale smooth optimization problems, and the effectiveness of the full BFGS method in solving small to medium-sized nonsmooth optimization problems, based on using a gradient, not a subgradient, oracle paradigm. We first summarize our theoretical results on the behavior of the scaled L-BFGS method with one update applied to a simple convex nonsmooth function that is unbounded below, stating conditions under which the method converges to a non-optimal point regardless of the starting point. We then turn to empirically investigating whether the same phenomenon holds more generally,focusing on a difficult problem of Nesterov, as well as eigenvalue optimization problems arising in semidefinite programming applications. We find that when applied to a nonsmooth function directly, L-BFGS, especially its scaled variant, often breaks down with a poor approximation to an optimal solution, in sharp contrast to full BFGS. Unscaled L-BFGS is less prone to breakdown but conducts far more function evaluations per iteration than scaled L-BFGS does, and thus it is slow. Nonetheless, it is often the case that both variants obtain better results than the provably convergent, but slow, subgradient method. On the other hand, when applied to Nesterov's smooth approximation of a nonsmooth function, scaled L-BFGS is generally much more efficient than unscaled L-BFGS, often obtaining good results even when the problem is quite ill-conditioned. Summarizing, for large-scale nonsmooth optimization problems for which full BFGS and other methods for nonsmooth optimization are not practical, it is often better to apply L-BFGS to a smooth approximation of a nonsmooth problem than to apply it directly to the nonsmooth problem.