We develop new variants of Benders decomposition methods for variational inequality problems. The construction is done by applying the general class of Dantzig–Wolfe decomposition of Luna et al. (Math Program 143(1–2):177–209, 2014) to an appropriately defined dual of the given variational inequality, and then passing back to the primal space. As compared to previous decomposition techniques of the Benders kind for variational inequalities, the following improvements are obtained. Instead of rather specific single-valued monotone mappings, the framework includes a rather broad class of multi-valued maximally monotone ones, and single-valued nonmonotone. Subproblems’ solvability is guaranteed instead of assumed, and approximations of the subproblems’ mapping are allowed (which may lead, in particular, to further decomposition of subproblems, which may otherwise be not possible). In addition, with a certain suitably chosen approximation, variational inequality subproblems become simple bound-constrained optimization problems, thus easier to solve.

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关于变分不等式的一类Benders分解方法

我们为变分不等式问题开发了Benders分解方法的新变体。通过应用Luna等人的Dantzig-Wolfe分解的一般类来完成构造。（数学程序143（1-2）：177-209，2014年）到给定变分不等式的适当定义对偶，然后传回原始空间。与先前针对变分不等式的Benders类分解技术相比，获得了以下改进。该框架包括相当广泛的一类多值最大单调映射和单值非单调映射，而不是相当具体的单值单调映射。子问题的可解性得到了保证，而不是假定的问题，并且允许子问题映射的近似值（尤其可能导致子问题的进一步分解，否则可能无法实现）。此外，具有一定的适当选择的近似，变分不等式子问题变得简单束缚约束优化问题，因而更容易解决。