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Radius Theorems for Monotone Mappings

Article

Abstract

For a Hilbert space X and a mapping \(F: X\rightrightarrows X\) (potentially set-valued) that is maximal monotone locally around a pair \((\bar {x},\bar {y})\) in its graph, we obtain a radius theorem of the following kind: the infimum of the norm of a linear and bounded single-valued mapping B such that F + B is not locally monotone around \((\bar {x},\bar {y}+B\bar {x})\) equals the monotonicity modulus of F. Moreover, the infimum is not changed if taken with respect to B symmetric, negative semidefinite and of rank one, and also not changed if taken with respect to all functions f : XX that are Lipschitz continuous around \(\bar {x}\) and ∥B∥ is replaced by the Lipschitz modulus of f at \(\bar {x}\). As applications, a radius theorem is obtained for the strong second-order sufficient optimality condition of an optimization problem, which in turn yields a lower bound for the radius of quadratic convergence of the smooth and semismooth versions of the Newton method. Finally, a radius theorem is derived for mappings that are merely hypomonotone.

Keywords

Monotone mappings Maximal monotone Locally monotone Radius theorem Optimization problem Second-order sufficient optimality condition Newton method 

Mathematics Subject Classification (2010)

47H05 49J53 90C31 

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematical ReviewsAnn ArborUSA
  2. 2.RMIT UniversityMelbourneAustralia
  3. 3.Department of MathematicsUniversity of WashingtonSeattleUSA

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