Abstract
In this paper, we propose an inertial derivative-free projection method for solving convex constrained nonlinear monotone operator equations (CNME). The method incorporates the inertial step with an existing method called derivative-free projection (DFPI) method for solving CNME. The reason is to improve the convergence speed of DFPI as it has been shown and reported in several works that indeed the inertial step can speed up convergence. The global convergence of the proposed method is proved under some mild assumptions. Finally, numerical results reported clearly show that the proposed method is more efficient than the DFPI.
Highlights
C ONSIDER the problem of finding y ∈ E such that T (y) = 0, (1)where T : Rn → Rn is a monotone and Lipschitz continuous operator and E is a nonempty, closed and convex subset of Rn.This problem has recently received remarkable attention as it arise in a number of applicable problems
Inspired by the inertial methods [36]–[41] and the derivative-free projection method proposed by Sun and Liu [17] which is an extension of the work of Cheng [42], we propose an inertial derivative-free projection method for finding solutions to problem (1)
Observe that as the parameter θk moves close to zero the convergence is slow and when θk approaches 1 but not equal to 1 the convergence is faster Based on the Sun and Liu [17] derivative-free projection method for monotone nonlinear equation with convex constraints called DFPI, we present an inertial derivative-free projection method for finding solutions to problem (1)
Summary
Where T : Rn → Rn is a monotone and Lipschitz continuous operator and E is a nonempty, closed and convex subset of Rn.This problem has recently received remarkable attention as it arise in a number of applicable problems. Recently several derivative-free methods such as the conjugate gradient (CG) method have been proposed for solving problem (1). Several researchers are interested in how to improve the speed of convergence of existing iterative algorithms. One of the approach in this regard is the inertial extrapolation where a new step called the inertial step is added to the existing step(s) of an iterative method. It has been shown that the inertial step enhance the speed of the existing methods such as methods for solving fixed point problems, variational inequality problems, equilibrium problems, split feasibility problems, and so on. Inertial extrapolation method has been employed successfully in improving the convergence of the sequence generated by an inertial algorithm. The method is based on the work of Sun and Liu [17], where the inertial term is incorporated in order speed up its convergence. The symbol · stands for Euclidean norm on Rn
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