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

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 θ_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 ‖F(xk+αk′dk)‖ was not established by the authors, instead the authors used the bound for ‖ F(x_k+ α_kd_k) ‖ as the bound for ‖F(xk+αk′dk)‖ . In this paper, We first describe the necessary adjustments and establish the bound for ‖F(xk+αk′dk)‖ , after which the proposed approach by Liu and Feng continues to converge globally. In addition, we provide some numerical results to support the adjustments.