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The purpose of this paper is to present accelerations of the Mann and CQ algorithms. We first apply the Picard algorithm to the smooth convex minimization problem and point out that the Picard algorithm is the steepest descent method for solving the minimization problem. Next, we provide the accelerated Picard algorithm by using the ideas of conjugate gradient methods that accelerate the steepest descent method. Then, based on the accelerated Picard algorithm, we present accelerations of the Mann and CQ algorithms. Under certain assumptions, we show that the new algorithms converge to a fixed point of a nonexpansive mapping. Finally, we show the efficiency of the accelerated Mann algorithm by numerically comparing with the Mann algorithm. A numerical example is provided to illustrate that the acceleration of the CQ algorithm is ineffective.

Accelerated Mann and CQ algorithms for finding a fixed point of a nonexpansive mapping