ABSTRACT

In this paper, we propose modified Tseng's extragradient methods with self-adaptive step size for solving a bilevel split variational inequality problem (BSVIP) involving a strongly monotone mapping in the upper-level problem and pseudomonotone mappings in the lower-level one. This algorithm is very simple in the sense that it requires only two projections at each iteration step. The strong convergence of the proposed algorithm is established without the prior knowledge of the Lipschitz and strongly monotone constants of the mappings. In addition, the implementation of the method does not require the computation or estimation of the norm of the given operator, which is in general not an easy work in practice. Special cases are considered. Finally, a numerical example is given to illustrate the performance of the proposed algorithm in comparison with a previously known the subgradient extragradient algorithm.

摘要

在本文中，我们提出了改进的具有自适应步长的 Tseng 外梯度方法，用于解决涉及上层问题中的强单调映射和下层问题中的伪单调映射的双层分裂变分不等式问题 (BSVIP)。该算法非常简单，因为它在每个迭代步骤中只需要两个投影。所提出算法的强收敛性是在没有 Lipschitz 的先验知识和映射的强单调常数的情况下建立的。此外，该方法的实现不需要计算或估计给定算子的范数，这在实践中通常不是一件容易的工作。考虑特殊情况。最后，