The purpose of this paper is to introduce a new modified subgradient extragradient method for finding an element in the set of solutions of the variational inequality problem for a pseudomonotone and Lipschitz continuous mapping in real Hilbert spaces. It is well known that for the existing subgradient extragradient methods, the step size requires the line-search process or the knowledge of the Lipschitz constant of the mapping, which restrict the applications of the method. To overcome this barrier, in this work we present a modified subgradient extragradient method with adaptive stepsizes and do not require extra projection or value of the mapping. The advantages of the proposed method only use one projection to compute and the strong convergence proved without the prior knowledge of the Lipschitz constant of the inequality variational mapping. Numerical experiments illustrate the performances of our new algorithm and provide a comparison with related algorithms.

本文的目的是介绍一种新的改进的次梯度超梯度方法，该方法用于在实希尔伯特空间中拟单调和Lipschitz连续映射的变分不等式问题的解集中找到一个元素。众所周知，对于现有的次梯度超梯度方法，步长需要线搜索过程或映射的Lipschitz常数的知识，这限制了该方法的应用。为了克服这一障碍，在这项工作中，我们提出了一种具有自适应步长的改进的次梯度超梯度方法，不需要额外的投影或映射值。该方法的优点是仅使用一个投影进行计算，并且在不了解不等式变分映射的Lipschitz常数的情况下证明了强收敛性。