Abstract The projection methods with vanilla inertial extrapolation step for variational inequalities have been of interest to many authors recently due to the improved convergence speed contributed by the presence of inertial extrapolation step. However, it is discovered that these projection methods with inertial steps lose the Fejer monotonicity of the iterates with respect to the solution, which is being enjoyed by their corresponding non-inertial projection methods for variational inequalities. This lack of Fejer monotonicity makes projection methods with vanilla inertial extrapolation step for variational inequalities not to converge faster than their corresponding non-inertial projection methods at times. Also, it has recently been proved that the projection methods with vanilla inertial extrapolation step may provide convergence rates that are worse than the classical projected gradient methods for strongly convex functions. In this paper, we introduce projection methods with alternated inertial extrapolation step for solving variational inequalities. We show that the sequence of iterates generated by our methods converges weakly to a solution of the variational inequality under some appropriate conditions. The Fejer monotonicity of even subsequence is recovered in these methods and linear rate of convergence is obtained. The numerical implementations of our methods compared with some other inertial projection methods show that our method is more efficient and outperforms some of these inertial projection methods.

中文翻译：

---

变分不等式的交替惯性步长投影方法：弱收敛和线性收敛

摘要 由于惯性外推步骤的存在提高了收敛速度，最近许多作者对具有原始惯性外推步骤的变分不等式投影方法感兴趣。然而，发现这些具有惯性步骤的投影方法失去了迭代关于解的费耶单调性，这是它们对应的变分不等式的非惯性投影方法所享有的。这种 Fejer 单调性的缺乏使得具有用于变分不等式的普通惯性外推步骤的投影方法有时不会比其相应的非惯性投影方法收敛得更快。还，最近已经证明，具有普通惯性外推步骤的投影方法可以提供比强凸函数的经典投影梯度方法更差的收敛速度。在本文中，我们介绍了具有交替惯性外推步骤的投影方法，用于解决变分不等式。我们表明，在某些适当条件下，我们的方法生成的迭代序列弱收敛于变分不等式的解。这些方法恢复了偶子序列的费耶单调性，并获得了线性收敛速度。与其他一些惯性投影方法相比，我们方法的数值实现表明我们的方法更有效，并且优于其中一些惯性投影方法。