The purpose of this article is to introduce a general inertial projected gradient method with a self-adaptive stepsize for solving variational inequality problems. The proposed method incorporates two different extrapolations with respect to the previous iterates into the projected gradient method. The weak convergence for our method is proved under standard assumptions without any requirement of the knowledge of the Lipschitz constant of the mapping. Furthermore, R-linear convergence rate is established under the strong monotonicity assumption of the mapping. Finally preliminary results from numerical experiments and applications to optimal control problems are performed which show the advantage of the proposed method.

本文的目的是介绍一种具有自适应步长的通用惯性投影梯度方法，用于解决变分不等式问题。所提出的方法将相对于先前迭代的两种不同的外推结合到投影梯度方法中。我们方法的弱收敛性是在标准假设下证明的，无需了解映射的 Lipschitz 常数。此外，R-线性收敛速度是在映射的强单调性假设下建立的。最后从数值实验和应用到最优控制问题的初步结果显示了所提出的方法的优点。