论文标题
使用切换约束的数学计划的拓扑方法
Topological approach to mathematicalprograms with switching constraints
论文作者
论文摘要
我们从拓扑角度研究了具有切换约束（MPSC）的数学程序。证明了莫尔斯理论的两个基本定理。在W站点集之外，可以执行较低级别集的连续Defor-Sation。但是，当通过W站立水平时，较低级别集的拓扑通过附着W维单元而变化。尺寸w等于非等级W-stationary点的W-索引。 W-指数既取决于受限拉格朗日的Hessian的负征值的数量，又取决于双活动开关约束的数量。结果，我们显示了MPSC的山间通过定理。此外，我们解决了以下问题，在MPSC的背景下，W-Stationary点的非确定性是否过于限制。事实证明，所有W-安置点都是非排定的。此外，我们研究了W-Staration Points的非排步和弦稳定性之间的差距。提供了通过线性独立约束资格下定义函数的第一阶和二阶信息对W-站点的完整表征。特别是，强稳定的W-安置点的所有双活动拉格朗日乘数都不会消失。
We study mathematical programs with switching constraints (MPSC)from the topological perspective. Two basic theorems from Morse theory are proved. Outside the W-stationary point set, continuous defor-mation of lower level sets can be performed. However, when passing a W-stationary level, the topology of the lower level set changes via the attachment of a w-dimensional cell. The dimension w equals the W-index of the nondegenerate W-stationary point. The W-index depends on both the number of negative eigenvalues of the restricted Lagrangian's Hessian and the number of bi-active switching constraints. As a consequence, we show the mountain pass theorem for MPSC. Additionally, we address the question if the assumption on the nondegeneracy of W-stationary points is too restrictive in the context of MPSC. It turns out that all W-stationary points are generically nondegenerate. Besides, we examine the gap between nondegeneracy andstrong stability of W-stationary points. A complete characterizationof strong stability for W-stationary points by means of first and second order information of the MPSC defining functions under linear independence constraint qualification is provided. In particular, all bi-active Lagrange multipliers of a strongly stable W-stationary point cannot vanish.
