Abstract
In recent times, various algorithms have been incorporated with the inertial extrapolation step to speed up the convergence of the sequence generated by these algorithms. As far as we know, very few results exist regarding algorithms of the inertial derivative-free projection method for solving convex constrained monotone nonlinear equations. In this article, the convergence analysis of a derivative-free iterative algorithm (Liu and Feng in Numer. Algorithms 82(1):245–262, 2019) with an inertial extrapolation step for solving large scale convex constrained monotone nonlinear equations is studied. The proposed method generates a sufficient descent direction at each iteration. Under some mild assumptions, the global convergence of the sequence generated by the proposed method is established. Furthermore, some experimental results are presented to support the theoretical analysis of the proposed method.
Highlights
Our main aim in this paper is to find the approximate solutions of the systems of monotone nonlinear equations with convex constraints; precisely, the problem find x ∈ C s.t. h(x) = 0, (1)
Numerous iterative methods have been proposed by many authors to approximate solutions of (1)
Our proposed method is a combination of inertial extrapolation step and the derivative-free iterative method for nonlinear monotone equations with convex constraints [1]
Summary
Numerous iterative methods have been proposed by many authors to approximate solutions of (1) (see [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35] and the references therein). Equation (3) yields the following iterative algorithm: xk+1 = xk + β(xk – xk–1) – α∇f (xk), k ≥ 1, (4) Where β = 1 – γ j, α = j2 and β(xk – xk–1) is called the inertial extrapolation term which is intended to speed up the convergence of the sequence generated by equation (4).
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