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

摘要

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 ?(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 IIF(xk +a?kdk)II was not established by the authors, instead the authors used the bound for IIF(x(k) +a(k)d(k))II as the bound for IIF(x(k )+a(k)d(k))II. In this paper, We first describe the necessary adjustments and establish the bound for IIF(x(k) + a(k)d(k))II, after which the proposed approach by Liu and Feng continues to converge globally. In addition, we provide some numerical results to support the adjustments.

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