Radius Theorems for Monotone Mappings

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 : X → X 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.