Multivariate Monotone Inclusions in Saddle Form.

We introduce the notion of a saddle operator for highly structured multivariate monotone inclusions involving a mix of set-valued, cocoercive, and Lipschitzian monotone operators, as well as various monotonicity-preserving operations among them. The properties of this saddle operator are investigated, and an asynchronous block-iterative algorithm to find its zeros is designed and analyzed. In turn, this allows us to solve the original system via a novel splitting algorithm of great flexibility in terms of processing the constituent operators individually and exploiting their specific attributes. The case of multivariate minimization problems is discussed.