统一分离条件：光滑与非光滑优化的一般约束规范

中文摘要 AI 辅助

我们引入了统一分离条件（USC）——一个用于有限维非线性规划的简单而一般的约束规范。USC要求原点在可行集的邻域内与约束违反函数的次微分均匀分离。本文的主要结果有两个方面。首先，我们证明了经典的不等式Mangasarian--Fromovitz约束规范（MFCQ）和等式的线性独立约束规范（LICQ）蕴含USC。反之，USC严格更一般：它在经典条件未定义的非光滑环境中仍然适用，并且即使MFCQ失效时也可能成立。MFCQ蕴含USC的证明基于Gordan定理；LICQ到USC的蕴含关系则来自活动约束梯度的线性独立性。我们还引入了USC的局部版本，并讨论了通过凸二次规划进行计算验证的方法。USC为约束规范提供了一个统一框架，连接了经典光滑理论与非光滑优化。

英文摘要

We introduce the Unified Separation Condition (USC) - a simple and general constraint qualification for finite-dimensional nonlinear programming. The USC requires that the origin be uniformly separated from the subdifferential of the constraint violation function in a neighborhood of the feasible set. The main result of this paper is two-fold. First, we show that the classical Mangasarian--Fromovitz constraint qualification (MFCQ) for inequalities and the linear independence constraint qualification (LICQ) for equalities imply the USC. Conversely, USC is strictly more general: it remains applicable in nonsmooth settings where classical conditions are not defined, and it may hold even when MFCQ fails. The proof that MFCQ implies USC is based on Gordan's theorem; the implication from LICQ to USC follows from the linear independence of the gradients of the active constraints. We also introduce a local version of USC and discuss its computational verification via convex quadratic programming. The USC provides a unified framework for constraint qualifications, bridging classical smooth theory and nonsmooth optimization.

发表机构

• St. Petersburg State University（圣彼得堡国立大学）

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