Analysis of Limited-Memory BFGS on a Class of Nonsmooth Convex Functions

The limited memory BFGS (L-BFGS) method is widely used for large-scale unconstrained optimization, but its behavior on nonsmooth problems has received little attention. L-BFGS can be used with or without "scaling"; the use of scaling is normally recommended. A simple special case, when just one BFGS update is stored and used at every iteration, is sometimes also known as memoryless BFGS. We analyze memoryless BFGS with scaling, using any Armijo-Wolfe line search, on the function $f(x) = a|x^{(1)}| + \sum_{i=2}^{n} x^{(i)}$, initiated at any point $x_0$ with $x_0^{(1)}\not = 0$. We show that if $a\ge 2\sqrt{n-1}$, the absolute value of the normalized search direction generated by this method converges to a constant vector, and if, in addition, $a$ is larger than a quantity that depends on the Armijo parameter, then the iterates converge to a non-optimal point $\bar x$ with $\bar x^{(1)}=0$, although $f$ is unbounded below. As we showed in previous work, the gradient method with any Armijo-Wolfe line search also fails on the same function if $a\geq \sqrt{n-1}$ and $a$ is larger than another quantity depending on the Armijo parameter, but scaled memoryless BFGS fails under a weaker condition relating $a$ to the Armijo parameter than that implying failure of the gradient method. Furthermore, in sharp contrast to the gradient method, if a specific standard Armijo-Wolfe bracketing line search is used, scaled memoryless BFGS fails when $a\ge 2 \sqrt{n-1}$ regardless of the Armijo parameter. Finally, numerical experiments indicate that similar results hold for scaled L-BFGS with any fixed number of updates.