Resolvents of Monotone Operators

Abstract

Two quite useful single-valued, Lipschitz continuous operators can be associated with a monotone operator, namely its resolvent and its Yosida approximation. This chapter is devoted to the investigation of these operators. It exemplifies the tight interplay between firmly nonexpansive mappings and monotone operators. Indeed, firmly nonexpansive operators with full domain can be identified with maximally monotone operators via resolvents and the Minty parametrization. When specialized to subdifferential operators, resolvents become proximity operators. Numerous calculus rules for resolvents are derived. Finally, we address the problem of finding a zero of a maximally monotone operator, via the proximal-point algorithm and via approximating curves.