Various versions of inertial subgradient extragradient methods for solving variational inequalities have been and continue to be studied extensively in the literature. In many of the versions that were proposed and studied, the inertial factor, which speeds up the convergence of the method, is assumed to be less than 1, and in many cases, stringent conditions are also required in order to obtain convergence. Several of the conditions assumed in the literature make the proposed inertial subgradient extragradient method computationally difficult to implement in some cases. In the present paper, we investigate the subgradient extragradient algorithm for solving variational inequality problems in real Hilbert spaces and consider it with inertial extrapolation terms and self-adaptive step sizes. We present a relaxed version of this method with seemingly easier to implement conditions on the inertial factor and the relaxation parameter. In the method we propose, the inertial factor can be chosen in a special case to be 1, a choice which is not possible in the inertial subgradient extragradient methods proposed in the literature. We also provide some numerical examples which illustrate the effectiveness and competitiveness of our algorithm.

用于解决变分不等式的惯性次梯度超梯度方法的各种版本已经并且继续在文献中进行广泛研究。在许多已提议和研究的版本中，假定加速该方法收敛的惯性因子小于1，并且在许多情况下，还需要严格的条件才能获得收敛。文献中假设的几种条件使所提出的惯性次梯度超梯度方法在某些情况下难以实现。在本文中，我们研究了解决实际希尔伯特空间中的变分不等式问题的次梯度超梯度算法，并考虑了惯性外推项和自适应步长。我们提出了这种方法的宽松版本，在惯性因子和松弛参数上似乎更容易实现。在我们提出的方法中，在特殊情况下可以将惯性因子选择为1，这在文献中提出的惯性次梯度超梯度方法中是不可能的。我们还提供了一些数值示例，说明了我们算法的有效性和竞争力。