We discuss the convergence of regularization methods for mathematical programs with complementarity constraints with approximate sequence of stationary points. It is now well accepted in the literature that, under some tailored constraint qualification, the genuine necessary optimality condition for this problem is the M-stationarity condition. It has been pointed out, (Kanzow and Schwartz in Math Oper Res 40(2):253–275. 2015), that relaxation methods with approximate stationary points fail to ensure convergence to M-stationary points. We define a new strong approximate stationarity concept, and we prove that a sequence of strong approximate stationary points always converges to an M-stationary solution. We also prove under weak assumptions the existence of strong approximate stationary points in the neighborhood of an M-stationary solution.

中文翻译：

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具有互补约束的正则化数学规划的近似平稳点

我们讨论了具有近似静止点序列的互补约束的数学程序的正则化方法的收敛性。现在在文献中被广泛接受的是，在一些定制的约束条件下，这个问题的真正必要的最优条件是 M 平稳条件。已经指出（Kanzow 和 Schwartz 在 Math Oper Res 40(2):253–275. 2015 中），具有近似静止点的松弛方法无法确保收敛到 M 静止点。我们定义了一个新的强近似平稳性概念，并证明了一系列强近似平稳点总是收敛到一个 M 平稳解。我们还在弱假设下证明了在 M 平稳解的邻域中存在强近似驻点。