Higher-order metric subregularity and its applications

Abstract

This paper is devoted to the study of metric subregularity and strong subregularity of any positive order q for set-valued mappings in finite and infinite dimensions. While these notions have been studied and applied earlier for q = 1 and-to a much lesser extentfor q ∈ (0, 1), no results are available for the case q > 1. We derive characterizations of these notions for subgradient mappings, develop their sensitivity analysis under small perturbations, and provide applications to the convergence rate of Newton-type methods for solving generalized equations.Keywords Variational analysis · Metric subregularity and strong subregularity of higher order · Newton and quasi-Newton methods · Generalized normals and subdifferentials Mathematics Subject Classification 49J52 · 90C30 · 90C31

Error Bounds and Hölder Metric Subregularity

“…The history of the Hölder metric subregularity property seems to be significantly shorter with most work done in the last few years; cf. [24,38,39,43,45,51,57]. Note the only attempt so far to consider the case q > 1 in [51].…”

Nonlinear Metric Subregularity

“…(ii) Substituting (i) into (32) and taking into account that D * F (x, y)(tv * ) = tD * F (x, y)(v * ) for any v * ∈ Y * and t > 0, we obtain: Similarly, substituting (i) into (33), we obtain (iii). ⊓ ⊔ Now we can define the strict subdifferential ϕ-slope, approximate strict subdifferential ϕ-slope, modified strict subdifferential ϕ-slope, and modified approximate strict subdifferential ϕ-slope of F at (x,ȳ):…”

“…They do not depend on ϕ. Using some simple calculus, one can formulate representations for (g, ρ)-slopes (32) and (33) in the case when g is given by (44). In the next proposition and the rest of the article, we use the notation…”

“…The history of the nonlinear/Hölder metric subregularity property seems to be significantly shorter with most work done in the last few years, cf. [29][30][31][32][33][34][35][36], although some studies of such properties can be found in earlier publications, cf. e.g.…”

Directional Hölder Metric Regularity

“…Apart from the study of the usual (Lipschitz) metric regularity, Hölder metric regularity or more generally nonlinear metric regularity have been studied over the years 1980 − 1990s by several authors, including for example Borwein and Zhuang [1], Frankowska [16], Penot [25], and recently, for instance, Frankowska and Quincampoix [17], Ioffe [26], Li and Mordukhovich [27], Oyang and Mordukovhich [28].…”