Global convergence of full BFGS under convexity and smoothness

Determine whether every full-BFGS sequence generated with an Armijo-weak-Wolfe line search for a convex, globally L-smooth function with a nonempty optimal solution set satisfies f(x_k)\to f^\star, or construct a single smooth convex counterexample.

Background

The paper establishes an unconditional best-iterate gradient guarantee of order O(k{-1/2}) for BFGS with an Armijo-Wolfe line search under convexity, global Lipschitz continuity of the gradient, and boundedness below. It also derives an O(k{-1}) function-gap rate under an additional boundedness condition on the initial sublevel set or an equivalent gap-to-gradient bound.

The unresolved issue is whether function values necessarily converge to the optimum under only convexity, global L-smoothness, and nonemptiness of the optimal solution set, without the additional bounded-sublevel-set assumption. The authors frame the alternative possibility as constructing a single smooth convex function for which a full-BFGS Armijo-weak-Wolfe sequence fails to converge optimally.

References

Two questions remain open. First, under only convexity, global $L$-smoothness, and $X\star\ne\varnothing$, must every full-BFGS Armijo-weak-Wolfe sequence satisfy $f(x_k)\to f\star$, or can one construct a single smooth convex counterexample?

— On the Complexity of BFGS Method for Smooth Convex Optimization  (2608.16009 - Ding et al., 17 Aug 2026) in Section 4, “Conclusion and open problems”