Splitting Compositions of Monotone Operators

Abstract

Coupled convex optimisation equilibria, bilinear saddle and potential systems, and nonlinear electronic device laws lead naturally to inclusions of the form $0\in Hx+A(Bx)-b$, where $A$ and $B$ are maximal monotone but their composition need not be monotone. We introduce a new class of splitting algorithms that solve such inclusions. Introducing the intermediate graph variable $y\in Bx$ exposes a directed two-row system. A one-parameter family of kernels produces a lower-triangular warped-resolvent iteration evaluated with one application of $H$ and the ordinary resolvents $J_{αA}$ and $J_{βB}$. Appropriate specializations of the problem data and algorithmic parameters recover forward--backward, inertial, primal--dual, and Douglas--Rachford constructions, while setting $H=b=0$ generates three novel algorithms for the pure composition $0\in A(Bx)$. Convergence of the coupled method is certified by an explicit nonnegative two-block gain matrix. Its Schur condition yields contraction in particular weighted norms, global linear convergence, and residual error estimates. For the general problem class, our convergence condition is shown to be sharp for an explicitly constructed linear family. When the problem is specialized to a pure composition, we prove convergence under the weaker assumption that only one operator is strongly monotone. This weaker regime is illustrated on a nonlinear circuit problem. We further illustrate our methods by solving inference problems for cascaded equilibrium neural networks.