In this paper, we introduce an efficient subgradient extragradient (SE) based method for solving variational inequality problems with monotone operator in Hilbert space. In many existing SE methods, two values of operator are needed over each iteration and the Lipschitz constant of the operator or linesearch is required for estimating step sizes, which are usually not practical and expensive. To overcome these drawbacks, we present an inertial SE based algorithm with adaptive step sizes, estimated by using an approximation of the local Lipschitz constant without running a linesearch. Each iteration of the method only requires a projection on the feasible set and a value of the operator. The numerical experiments illustrate the efficiency of the proposed algorithm.

在本文中，我们介绍了一种基于有效次梯度超梯度（SE）的方法，用于解决希尔伯特空间中单调算子的变分不等式问题。在许多现有的SE方法中，每次迭代都需要两个运算符值，并且需要运算符或Lipsearch的Lipschitz常数来估算步长，这通常不实用且昂贵。为了克服这些缺点，我们提出了一种具有自适应步长的基于惯性SE的算法，该算法通过使用局部Lipschitz常数的近似值进行估算而无需进行线搜索。该方法的每次迭代仅需要在可行集上的投影和算子的值。数值实验说明了该算法的有效性。