The Krasnosel'skii-Mann (KM) algorithm is the most fundamental iterative scheme designed to find a fixed point of an averaged operator in the framework of a real Hilbert space, since it lies at the heart of various numerical algorithms for solving monotone inclusions and convex optimization problems. We enhance the Krasnosel'skii-Mann algorithm with Nesterov's momentum updates and show that the resulting numerical method exhibits a convergence rate for the fixed point residual of $o(1/k)$ while preserving the weak convergence of the iterates to a fixed point of the operator. Numerical experiments illustrate the superiority of the resulting so-called Fast KM algorithm over various fixed point iterative schemes, and also its oscillatory behavior, which is a specific of Nesterov's momentum optimization algorithms.

翻译：Krasnosel'skii-Mann（KM）算法是实Hilbert空间框架下设计用于寻找平均算子不动点的最基础迭代方案，因其是求解单调包含问题和凸优化问题的多种数值算法的核心。我们通过引入Nesterov动量更新对Krasnosel'skii-Mann算法进行改进，并证明所得数值方法在保持迭代序列弱收敛至算子不动点的同时，定点残差具有$o(1/k)$的收敛率。数值实验表明，相较于各类定点迭代方案，所得所谓快速KM算法具有显著优越性，同时展现出Nesterov动量优化算法特有的振荡行为。