Robust Convergence Analysis of Three-Operator Splitting.

Operator splitting methods solve composite optimization problems by breaking them into smaller sub-problems that can be solved sequentially or in parallel. In this paper, we propose a unified framework for certifying both linear and sublinear convergence rates for three-operator splitting (TOS) method under a variety of assumptions about the objective function. By viewing the algorithm as a dynamical system with feedback uncertainty (the oracle model), we leverage robust control theory to analyze the worst-case performance of the algorithm using matrix inequalities. We then show how these matrix inequalities can be used to guide the search for selecting the parameters of the algorithm (both symbolically and numerically) for optimal worst-case performance. Our framework yields tighter bounds relative to competing bounds in the literature. We illustrate our results numerically by solving an input-constrained optimal control problem.