The paper is mainly devoted to systematic developments and applications of geometric aspects of second-order variational analysis that are revolved around the concept of parabolic regularity of sets. This concept has been known in variational analysis for more than two decades while being largely underinvestigated. We discover here that parabolic regularity is the key to derive new calculus rules and computation formulas for major second-order generalized differential constructions of variational analysis in connection with some properties of sets that go back to classical differential geometry and geometric measure theory. The established results of second-order variational analysis and generalized differentiation, being married to the developed calculus of parabolic regularity, allow us to obtain novel applications to both qualitative and quantitative/numerical aspects of constrained optimization including second-order optimality conditions, augmented Lagrangians, etc. under weak constraint qualifications.

中文翻译：

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几何变分分析的抛物线规律

本文主要致力于围绕集合抛物线正则性概念的二阶变分分析的几何方面的系统发展和应用。这个概念在变分分析中已经为人所知 20 多年，但在很大程度上没有得到充分研究。我们在这里发现，抛物线规律性是为变分分析的主要二阶广义微分构造推导出新的微积分规则和计算公式的关键，这些构造与可以追溯到经典微分几何和几何测度理论的集合的某些性质有关。二阶变分分析和广义微分的既定结果，与已发展的抛物线规律性微积分相结合，