# 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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## Authors and Affiliations

• A. L. Dontchev
• 1
• A. Eberhard
• 2
• R. T. Rockafellar
• 3
1. 1.Mathematical ReviewsAnn ArborUSA
2. 2.RMIT UniversityMelbourneAustralia
3. 3.Department of MathematicsUniversity of WashingtonSeattleUSA