Monotone Inclusions, Acceleration, and Closed-Loop Control

We propose and analyze a new dynamical system with closed-loop control law in Hilbert space [Formula: see text], aiming to shed light on the acceleration phenomenon for monotone inclusion problems, which unifies broad class of optimization, saddle point, variational inequality (VI) problems under single framework. Given an operator text] that is maximal monotone, we governed by where feedback tuned resolution algebraic equation some text]. Our first contribution prove existence uniqueness global solution via Cauchy–Lipschitz theorem. present simple Lyapunov function establishing weak convergence trajectories Opial lemma strong results additional conditions. then ergodic rate terms gap pointwise residue function. Local linear established distance error bound condition. Further, provide algorithmic framework based implicit discretization our Euclidean setting, generalizing large-step hybrid proximal extragradient Even though discrete-time analysis simplification generalization existing analyses bounded domain, it largely motivated aforementioned continuous-time analysis, illustrating fundamental role plays inclusion. A highlight result concerning text]-order tensor algorithms complementing recent point VI problems. Funding: This work was supported part Mathematical Data Science Program Office Naval Research [Grant N00014-18-1-2764] Vannevar Bush Faculty Fellowship N00014-21-1-2941].