In this work, we investigate pseudomonotone variational inequality problems in a real Hilbert space and propose two projection-type methods with inertial terms for solving them. The first method does not require prior knowledge of the Lipschitz constant and the second one does not require the Lipschitz continuity of the mapping which governs the variational inequality. A weak convergence theorem for our first algorithm is established under pseudomonotonicity and Lipschitz continuity assumptions, and a weak convergence theorem for our second algorithm is proved under pseudomonotonicity and uniform continuity assumptions. We also establish a nonasymptotic O(1/n) convergence rate for our proposed methods. In order to illustrate the computational effectiveness of our algorithms, some numerical examples are also provided.

在这项工作中，我们研究了真实希尔伯特空间中的伪单调变分不等式问题，并提出了两种带有惯性项的投影型方法来求解它们。第一种方法不需要先验的Lipschitz常数知识，第二种方法不需要控制变分不等式的映射的Lipschitz连续性。在伪单调性和Lipschitz连续性假设下建立了我们第一种算法的弱收敛定理，在伪单调性和一致连续性假设下证明了我们第二种算法的弱收敛定理。我们还建立了一个非渐近的O（1 / n）的收敛速度。为了说明我们算法的计算效率，还提供了一些数值示例。