Function-gap rate based only on initial distance

Prove a function-gap convergence rate for full BFGS with an Armijo-Wolfe line search that depends only on the initial distance dist(x_0,X^\star), rather than on the boundedness quantity R_0 of the initial sublevel set.

Background

The paper proves an O(k{-1}) function-gap rate when the initial sublevel set is bounded, quantified by R_0, or when the gap-to-gradient quantity W_0 is finite. These quantities control the relationship between function gaps and gradient norms along the iterates.

The authors leave unresolved whether the dependence on the global initial-sublevel-set quantity R_0 can be replaced by dependence solely on the initial distance from x_0 to the optimal solution set X\star. Such a result would provide a more localized complexity guarantee.

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? Second, can one prove a function-gap rate depending only on the initial distance $dist(x_0,X\star)$, rather than on the boundedness quantity of the initial sublevel set $R_0$?

— 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”