The Iterates of the Frank-Wolfe Algorithm May Not Converge

The Frank-Wolfe algorithm is a popular method for minimizing a smooth convex function f over a compact convex set C. Whereas many convergence results have been derived in terms of function values, almost nothing is known about the convergence behavior of the sequence of iterates (xt)t is an element of N. Under the usual assumptions, we design several counterexamples to the convergence of (xt)t is an element of N, where f is d-time continuously differentiable, dP 2, and f(xt) -> minC f. Our counterexamples cover the cases of open-loop, closed-loop, and line-search step-size strategies and work for any choice of the linear minimization oracle, thus demonstrating the fundamental pathologies in the convergence behavior of (xt)t is an element of N.