  Several aspects of the interplay between monotone operator theory and convex
optimization are presented. The crucial role played by monotone operators in
the analysis and the numerical solution of convex minimization problems is
emphasized. We review the properties of subdifferentials as maximally monotone
operators and, in tandem, investigate those of proximity operators as
resolvents. In particular, we study new transformations which map proximity
operators to proximity operators, and establish connections with self-dual
classes of firmly nonexpansive operators. In addition, new insights and
developments are proposed on the algorithmic front.
