Parabolic Regularity in Geometric Variational Analysis

TL;DR

This paper explores the concept of parabolic regularity in geometric variational analysis, revealing its importance in deriving new calculus rules and applications for constrained optimization problems.

Contribution

It introduces the use of parabolic regularity to develop new calculus rules and formulas in second-order variational analysis, enhancing optimization methods.

Findings

New calculus rules for second-order generalized derivatives

Applications to second-order optimality conditions

Enhanced methods for constrained optimization

Abstract

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.