Generalized continuation Newton methods and the trust-region updating strategy for the underdetermined system

Both the paper and the model analyze the same generalized continuation Newton method x_{k+1} = x_k - Δt_k/(1+Δt_k) J_k^+ F_k with trust-region-style ratio ρ_k and update rule (24)–(25) . For global behavior, the paper proves lim inf_k ||F(x_k)|| = 0 (Theorem 1, eq. (40)) using Lemmas 1–2 and the subsequence-of-accepted-steps argument culminating in eq. (43) . The model's Phase 2 reproduces the same conclusion via the standard one-step residual bound and |1−ρ_k| ≤ (L/(2c_σ^2)) (Δt_k/(1+Δt_k)) ||F(x_k)|| (cf. eq. (38)), consistent with the paper’s estimate . The critical discrepancy lies in the local analysis. The paper derives (47) from (46), effectively replacing J_k^+J_k by I on ℝ^n and obtaining ||e_{k+1}|| ≤ (1/(1+Δt_k))||e_k|| + O(||e_k||^2) without projecting onto Range(J_k^T) . This step is invalid when m < n: J_k^+J_k is the orthogonal projector Π_k onto Range(J_k^T), not the identity. A simple counterexample F(x) = [x_1] shows the tangential component (in Null(J)) is unchanged, contradicting (47). The remainder of the paper’s local proof (including (61)) relies on this error . By contrast, the model correctly retains the projector: e_{k+1} = (I − α_k Π_k)e_k − α_k J_k^+R_k, concluding superlinear convergence to x* in the square case (m=n) or when e_0 ∈ Range(J(x*)^T), and only normal-component/residual superlinear decay otherwise. Hence the paper’s local superlinear-to-x* claim is generally false for underdetermined problems, while the model’s corrected conditions are right.