识别活动
Identifying Activity
AI总结：
本文在复合非光滑最小化框架下，研究约束优化中活动约束的识别问题，通过广义梯度的图分解给出近似临界点序列有限识别活动的条件，其中关键条件类似于Mangasarian-Fromovitz约束规格。
摘要：
从实践和理论角度来看，约束优化中活动约束的识别都很有意义，因为它有望在解的邻域内将不等式约束问题简化为等式约束问题。我们在更一般的复合非光滑最小化设置中研究这个问题，其中目标函数是光滑向量函数c与下半连续函数h（通常非光滑但结构良好）的复合。在此设置中，h的广义梯度的图通常可以分解为更简单子集的（非不相交）并集。“识别”即决定在给定解的临界性条件中哪些子集是“活动的”。我们给出了任何收敛的近似临界点序列有限识别活动的条件。这些性质中突出的是一个类似于Mangasarian-Fromovitz约束规格的条件，它确保了在解处满足最优性条件的乘子向量集合的有界性。
英文摘要：
Identification of active constraints in constrained optimization is of interest from both practical and theoretical viewpoints, as it holds the promise of reducing an inequality-constrained problem to an equality-constrained problem, in a neighborhood of a solution. We study this issue in the more general setting of composite nonsmooth minimization, in which the objective is a composition of a smooth vector function c with a lower semicontinuous function h, typically nonsmooth but structured. In this setting, the graph of the generalized gradient of h can often be decomposed into a union (nondisjoint) of simpler subsets. "Identification" amounts to deciding which subsets of the graph are "active" in the criticality conditions at a given solution. We give conditions under which any convergent sequence of approximate critical points finitely identifies the activity. Prominent among these properties is a condition akin to the Mangasarian-Fromovitz constraint qualification, which ensures boundedness of the set of multiplier vectors that satisfy the optimality conditions at the solution.
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