It is well known that the algorithms with using a proximal operator can be not convergent for monotone variational inequality problems in the general case. Malitsky (Optim. Methods Softw. 33 (1) 140–164, ??) proposed a proximal extrapolated gradient algorithm ensuring convergence for the problems, where the constraints are a finite-dimensional vector space. Based on this proximal extrapolated gradient techniques, we propose a new subgradient proximal iteration method for solving monotone multivalued variational inequality problems with the closed convex constraint. At each iteration, two strongly convex subprograms are required to solve separately by using proximal operators. Then, the algorithm is convergent for monotone and Lipschitz continuous cost mapping. We also use the proposed algorithm to solve a jointly constrained Cournot-Nash equilibirum model. Some numerical experiment and comparison results for convex nonlinear programming confirm efficiency of the proposed modification.

众所周知，在一般情况下，使用近端算子的算法不能收敛于单调变分不等式问题。Malitsky（最佳方法软件。33（1）140–164，??）提出了一种近端外推梯度算法，可确保问题的收敛性，其中约束是有限维向量空间。基于这种近端外推梯度技术，我们提出了一种新的次梯度近端迭代方法，用于求解具有封闭凸约束的单调多值变分不等式问题。在每次迭代中，需要两个强凸子程序来使用近端运算符分别求解。然后，该算法收敛于单调和Lipschitz连续成本映射。我们还使用提出的算法来求解联合约束的古诺-纳什均衡模型。凸非线性规划的一些数值实验和比较结果证实了所提出的修改的有效性。