General Resolvents for Monotone Operators: Characterization and Extension

Understand

Monotone operators, especially in the form of subdifferential operators, are of basic importance in optimization.

• It is well known since Minty, Rockafellar, and Bertsekas-Eckstein that in Hilbert space, monotone operators can be understood and analyzed from the alternative viewpoint of firmly nonexpansive mappings, which were found to be precisely the resolvents of monotone operators.
• For example, the proximal mappings in the sense of Moreau are precisely the resolvents of subdifferential operators.
• More general notions of "resolvent", "proximal mapping" and "firmly nonexpansive" have been studied.