Explicit continuation methods and preconditioning techniques for unconstrained optimization problems

The paper’s Theorem 1 proves lim inf ||∇f(x_k)|| = 0 for the Eptc method under Lipschitz gradient, strong convexity on a level set, and a trust-region-style time-step update. The proof hinges on: (i) a uniform lower bound on the model decrease via spectral lower bounds for the switching preconditioner (L-BFGS rank-two formula or exact Hessian inverse) and (ii) a uniform positive lower bound on Δt_k derived from |ρ_k−1| ≤ C Δt_k, which guarantees infinitely many accepted steps and a summability argument that forces lim inf ||∇f(x_k)|| = 0. These steps appear in the paper’s Algorithm 1 and Lemma 2/Lemma 3 leading to Theorem 1 (equations (13)–(15), (29)–(32), (38)–(41), and (42)–(44) in the PDF) . The candidate’s solution establishes the same result via a standard trust-region “expansion phase” contradiction: once Δt_k is small, |ρ_k−1| is small so steps are accepted and Δt_k grows geometrically; if ||∇f|| were bounded away from zero, each such phase (and, later, each accepted step with Δt_k above a threshold) would yield a uniform decrease, contradicting boundedness below. The candidate also derives uniform spectral bounds for H_k consistent with the paper’s L-BFGS update (equation (19)) and Hessian-inverse preconditioner, ensuring the needed model-decrease and ratio estimates . Net: both arguments are correct and compatible. Minor issues in the paper: (a) an indexing slip in Lemma 3’s concluding paragraph (ρ_{K−1} should be ρ_K) and (b) equation (43) appears to drop a square on ||g_{k_i}||, but the intended bound (with ||g||^2) is clear from Lemma 2 and suffices to conclude lim inf ||∇f|| = 0 .