Fast convergence of dynamical ADMM via time scaling of damped inertial dynamics

Hedy Attouch (1) , Zaki Chbani (2) , Jalal M. Fadili (3, 4, 5, 6) , Hassan Riahi (2)

1 UM - Université de Montpellier
2 UCA - Université Cadi Ayyad [Marrakech]
3 Equipe Image - Laboratoire GREYC - UMR6072
4 NU - Normandie Université
5 ENSICAEN - École Nationale Supérieure d'Ingénieurs de Caen
6 GREYC - Groupe de Recherche en Informatique, Image et Instrumentation de Caen

In this paper, we propose in a Hilbertian setting a second-order timecontinuous dynamic system with fast convergence guarantees to solve structured convex minimization problems with an affine constraint. The system is associated with the augmented Lagrangian formulation of the minimization problem. The corresponding dynamics brings into play three general time-varying parameters, each with specific properties, and which are respectively associated with viscous damping, extrapolation and temporal scaling. By appropriately adjusting these parameters, we develop a Lyapunov analysis which provides fast convergence properties of the values and of the feasibility gap. These results will naturally pave the way for developing corresponding accelerated ADMM algorithms, obtained by temporal discretization.

Mots clés: augmented Lagrangian; ADMM; damped inertial dynamics; convex constrained minimization; convergence rates; Lyapunov analysis; Nesterov accelerated gradient method

Domaines: Optimisation et contrôle [math.OC]; Traitement du signal et de l'image [eess.SP]; Systèmes dynamiques [math.DS]; Machine Learning [stat.ML]

Hedy Attouch, Zaki Chbani, Jalal M. Fadili, Hassan Riahi. Fast convergence of dynamical ADMM via time scaling of damped inertial dynamics. Journal of Optimization Theory and Applications, inPress, 193 (1-3), pp.704-736. ⟨10.1007/s10957-021-01859-2⟩. ⟨hal-03177752⟩