NEW Q-NEWTON’S METHOD MEETS BACKTRACKING LINE SEARCH: GOOD CONVERGENCE GUARANTEE, SADDLE POINTS AVOIDANCE, QUADRATIC RATE OF CONVERGENCE, AND EASY IMPLEMENTATION

The paper proves a global dichotomy and quadratic convergence for the New Q-Newton Backtracking method on Morse functions, via (i) a pigeonhole spectral-shift lemma, (ii) descent with Armijo backtracking, (iii) a compact-set step-size bound, and (iv) a projective-metric argument to force convergence when bounded; it then shows local stabilization near nondegenerate critical points so that the method reduces to the non-backtracking NQ-Newton in a neighborhood and inherits its local behavior, thereby avoiding saddles almost surely and achieving quadratic convergence at minima . The candidate solution establishes the same claims through a closely related—but not identical—line of reasoning: it proves the spectral-gap selection via the same pigeonhole argument, descent plus Armijo well-posedness using a Lipschitz gradient estimate, bounded-iterates ⇒ gradient→0, and a careful local linearization of the iteration map F(x)=x−|A(x)|^{-1}∇f(x) to obtain quadratic convergence at minima and a measure-zero stable set at saddles. The only substantive difference is that the paper’s global bifurcation (bounded ⇒ convergence) leverages a projective-metric device , whereas the model gives a high-level trapping argument; this is a minor gap in exposition rather than a contradiction. Overall, both are correct, with different proof organization and tools.