Parabolic Regularity in Geometric Variational Analysis

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.

Generalized Newton Algorithms for Tilt-Stable Minimizers in Nonsmooth Optimization

“…Proof. Since Θ is C 2 -cone reducible at f (x), the assumptions of Proposition 7.1(b) hold by [24,Theorem 6.2]. Furthermore, the nondegeneracy condition (7.6) yields the fulfillment of assumption (a) in Proposition 7.1.…”

“…As shown in Section 5, a constructive realization of Algorithm 5.1 for C 1,1 functions (and hence of its extension for prox-regular ones in Section 6) requires an additional assumption on the twice epi-differentiability of the cost function. To obtain efficient conditions for the twice epi-differentiability of the function ϕ from (7.1) in terms of the initial data of this problem, we invoke our recent developments [24] on parabolic regularity in second-order variational analysis.…”

“…Recall that a set Θ ⊂ IR m is parabolically regular at z ∈ Θ for ū ∈ IR m if whenever w ∈ IR m is such that d 2 δ Θ (z, ū)(w) < ∞ there exist, among the sequences t k ↓ 0 and w k → w with ∆ 2 t k δ Θ (z, ū)(w k ) → d 2 δ Θ (z, ū)(w), those with the additional property that lim sup k→∞ w k − w /t k < ∞ as k → ∞. This notion was introduced in [38, Definition 13.65], but was not explored in [38] or anywhere else (before the recent paper [24]) further than in the fully amenable setting. The paper [24] offers an extensive study of parabolic regularity by revealing its remarkable properties as an appropriate second-order regularity notion for a large class of sets that overwhelmingly appear in variational analysis and constrained optimization.…”

“…This notion was introduced in [38, Definition 13.65], but was not explored in [38] or anywhere else (before the recent paper [24]) further than in the fully amenable setting. The paper [24] offers an extensive study of parabolic regularity by revealing its remarkable properties as an appropriate second-order regularity notion for a large class of sets that overwhelmingly appear in variational analysis and constrained optimization. This class strictly includes all the C 2 -cone reducible sets in the sense of [2,Definition 3.135] and encompasses convex polyhedra, the second-order cone, the cone of symmetric and positive semidefinite matrices, etc.…”

“…Applying to such problems, the developed graphical derivative-based algorithm generates a new Newton-type algorithm involving second subderivatives of cost functions. Its justification and convergence analysis for tilt-stable minimizers employ the recent developments of [24] on parabolic regularity in second-order variational analysis. The rest of the paper is organized as follows.…”