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An induction theorem and nonlinear regularity models

Phan Q. Khanh (pqkhanhus***at***gmail.com)
Alexander Y. Kruger (a.kruger***at***federation.edu.au)
Nguyen H. Thao (hieuthaonguyen***at***students.federation.edu.au)

Abstract: A general nonlinear regularity model for a set-valued mapping $F:X\times\R_+\rightrightarrows Y$, where $X$ and $Y$ are metric spaces, is considered using special iteration procedures, going back to Banach, Schauder, Lusternik and Graves. Namely, we revise the \emph{induction theorem} from Khanh, \emph{J. Math. Anal. Appl.}, 118 (1986) and employ it to obtain basic estimates for studying regularity/openness properties. We also show that it can serve as a substitution of the Ekeland variational principle when establishing other regularity criteria. Then, we apply the induction theorem and the mentioned estimates to establish criteria for both global and local versions of regularity/openness properties for our model and demonstrate how the definitions and criteria translate into the conventional setting of a set-valued mapping $F:X\rightrightarrows Y$.

Keywords: metric regularity, induction theorem, Ekeland variational principle

Category 1: Convex and Nonsmooth Optimization

Citation: SIAM J. Optim., 2015, 25(4), 25612588. http://dx.doi.org/10.1137/140991157

Download: [PDF]

Entry Submitted: 10/11/2014
Entry Accepted: 10/11/2014
Entry Last Modified: 12/18/2015

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