In this paper we discuss the simple bilevel programming problem (SBP) and its extension, the simple mathematical programming problem under equilibrium constraints (SMPEC). Here we first define both these problems and study their interrelations. Next we study the various types of necessary and sufficient optimality conditions for the (SMPEC) problems, which occur under various reformulations. The optimality conditions for (SBP) are special cases of the results obtained for (SMPEC) when the lower level objective is the gradient of a convex function. Among the various optimality conditions presented in this article are the sequential optimality conditions, which do not need any constraint qualification. We also present a schematic algorithm for (SMPEC), where the sequential optimality conditions play a key role in the convergence analysis.

中文翻译：

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简单的双层编程和扩展

在本文中，我们讨论了简单的双层规划问题（SBP）及其扩展，即均衡约束下的简单数学规划问题（SMPEC）。在这里，我们首先定义这两个问题并研究它们的相互关系。接下来，我们研究了 (SMPEC) 问题的各种必要和充分最优条件，这些条件发生在各种重新表述下。(SBP) 的最优条件是 (SMPEC) 获得的结果的特殊情况，当低级目标是凸函数的梯度时。在本文提出的各种最优性条件中，有序列最优性条件，不需要任何约束条件。我们还提出了（SMPEC）的示意性算法，其中顺序最优性条件在收敛分析中起着关键作用。