@ARTICLE{AbsMahAnd2005,
author = "Absil, P.-A. and Mahony, R. and Andrews, B.",
title = "Convergence of the Iterates of Descent Methods for Analytic Cost Functions",
journal = "SIAM J. Optim.",
year = 2005,
volume = 6,
number = 2,
pages = "531--547",
}
A) In the first sentence of the proposition, replace "local minimum" by "global minimum". Then $\phi(x_K) \geq 0$ is guaranteed to hold in the final sentence of the proof, and the claim follows. (The proof can also be streamlined because $U_m$ is no longer relevant.)
B) Assume that the considered iteration (3.8) has the "vanishing steps" property at $x^*$ in the sense of Definition 3.2 of Rebjock and Boumal 2024 (doi:10.1007/s10107-024-02136-6). (As pointed out therein, this is a reasonable assumption.) In the proof, further require that $\epsilon > 0$ is small enough so that (i) $B_\epsilon(x^*)$ is included in the vanishing-steps neighborhood and (ii) the vanishing-step function $\eta$ satisfies $\eta(x) < \mathrm{dist}(\partial U_m,x^*) - \epsilon$ for all $x \in B_\epsilon(x^*)$. Then, when we reach the final sentence of the proof of Proposition 3.3, we have that $\|x_K - x^*\| \leq \|x_K - x_{K-1}\| + \|x_{K-1} - x^*\| < \eta(x_{K-1}) + \epsilon < \mathrm{dist}(\partial U_m,x^*) - \epsilon + \epsilon = \mathrm{dist}(\partial U_m,x^*)$, hence $x_K$ is in $U_m$, meaning that $\phi(x_K) \geq 0$. Hence, as in fix A) above, we have that $\phi(x_K) \geq 0$ is guaranteed to hold in the final sentence of the proof, and the claim follows.