Universal objective for all line-search parameters

Determine whether there exists a single smooth function bounded below on which every method in the conjugacy class \(\mathcal{C}\) fails to converge for all line-search parameter pairs \(0<c_1<c_2<1\) simultaneously.

Background

The paper constructs, for each prescribed pair of line-search parameters $(c_1,c_2)$ with $0<c_1<c_2<1$, a parameter-specific function $f\in C^\infty(\mathbb{R}^2)$ that is bounded below and has Lipschitz continuous gradient. On this function, every method in the conjugacy class $\mathcal{C}$—including full-memory BFGS, L-BFGS, the Broyden positive family, and the Hestenes–Stiefel conjugate-gradient method—generates a nonconvergent sequence under the corresponding first-local-minimizer line search.

The unresolved issue is whether the dependence of the constructed objective on $(c_1,c_2)$ can be eliminated: namely, whether one universal smooth, lower-bounded objective can force nonconvergence for the entire parameter range simultaneously.