Radius Theorems for Monotone Mappings

For a Hilbert space X and a mapping F: X⇉ X (potentially set-valued) that is maximal monotone locally around a pair (x̅,y̅) 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 (x̅,y̅+Bx̅) 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 x̅ and ∥ B ∥ is replaced by the Lipschitz modulus of f at x̅ . 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.