A Primer on Monotone Operator Methods

This tutorial paper presents the basic notation and results of monotone operators and operator splitting methods, with a focus on convex optimization. A very wide variety of algorithms, ranging from classical to recently developed, can be derived in a uniform way. The approach is to pose the original problem to be solved as one of finding a zero of an appropriate monotone operator; this problem in turn is then posed as one of finding a fixed point of a related operator, which is done using the fixed point iteration. A few basic convergence results then tell us conditions under which the method converges, and, in some cases, how fast. This approach can be traced back to the 1960s and 1970s, and is still an active area of research. This primer is a self-contained gentle introduction to the topic.