Some iterative algorithms on Riemannian manifolds and Banach spaces with good global convergence guarantee

The paper establishes: (i) termination/strict descent and criticality of cluster points for (Local) Backtracking GD and Backtracking New Q-Newton; (ii) convergence when the set of critical points is at most countable via a connected-limit-set/capture argument; (iii) KŁ-based convergence for first-order methods; and (iv) almost-sure avoidance of generalized saddles for Local Backtracking GD under C2 near saddles. For New Q-Newton and its backtracking variant on Riemannian manifolds, the paper proves only local saddle avoidance in general, and global (measure-zero) avoidance requires an additional Real analytic-like Strong local retraction assumption; without this assumption global avoidance is left for future work. See Theorem 1.2 for the global claims summary, Theorem 2.19 for convergence/criticality, Theorem 2.20 for LB-GD avoidance, and Theorem 2.21 plus Definition 2.13 for the Real analytic-like requirement for New Q-Newton. The candidate solution omits this extra hypothesis and claims global saddle avoidance for (Backtracking) New Q-Newton under mere C3 near saddles, which overstates the paper’s result on general manifolds. Hence, while most parts align, the New Q-Newton global avoidance claim is incorrect relative to the paper’s scope (Theorem 1.2 and Theorems 2.19–2.21; Definition 2.3 and Definition 2.9). Citations: Theorem 1.2 ; Strong local retraction/Local BT-GD definitions , ; BT-GD convergence/criticality (Theorem 2.19) ; LB-GD avoidance (Theorem 2.20) ; Real analytic-like retraction (Definition 2.13) and its necessity for New Q-Newton (Theorem 2.21) , .