Comment on: “A derivative-free iterative method for nonlinear monotone equations with convex constraints”

Abstract

For solving nonlinear monotone equations with convex constraints, Liu and Feng (Numer. Algoritm. 82(1):245–262, 2019) suggested a derivative-free iterative technique. Although they assert that the direction $d_k$ satisfies inequality (2.1), however, this is not true, as the derivation of the parameter $\theta _k$ given by equation (2.7) is not correct. This led to Lemma 2.2, Lemma 3.1 and Theorem 3.1 in Liu and Feng (Numer. Algoritm. 82(1):245–262, 2019) not holding. In addition, Theorem 3.1 is still invalid as the bound for $\Vert F(x_k+\alpha _k^{\prime }d_k)\Vert$ was not established by the authors, instead the authors used the bound for $\Vert F(x_k+\alpha _kd_k)\Vert$ as the bound for $\Vert F(x_k+\alpha _k^{\prime }d_k)\Vert$. In this paper, We first describe the necessary adjustments and establish the bound for $\Vert F(x_k+\alpha _k^{\prime }d_k)\Vert$, after which the proposed approach by Liu and Feng continues to converge globally. In addition, we provide some numerical results to support the adjustments.