Lyapunov Analysis For Monotonically Forward-Backward Accelerated Algorithms

Nesterov's accelerated gradient method (NAG) achieves faster convergence than gradient descent for convex optimization but lacks monotonicity in function values. To address this, Beck and Teboulle [2009b] proposed a monotonic variant, M-NAG, and extended it to the proximal setting as M-FISTA for composite problems such as Lasso. However, establishing the linear convergence of M-NAG and M-FISTA under strong convexity remains an open problem. In this paper, we analyze M-NAG via the implicit-velocity phase representation and show that an additional assumption, either the position update or the phase-coupling relation, is necessary to fully recover the NAG iterates. The essence of M-NAG lies in controlling an auxiliary sequence to enforce non-increase. We further demonstrate that the M-NAG update alone is sufficient to construct a Lyapunov function guaranteeing linear convergence, without relying on full NAG iterates. By modifying the mixed sequence to incorporate forward-indexed gradients, we develop a new Lyapunov function that removes the kinetic energy term, enabling a direct extension to M-NAG. The required starting index depends only on the momentum parameter and not on problem constants. Finally, leveraging newly developed proximal inequalities, we extend our results to M-FISTA, establishing its linear convergence and deepening the theoretical understanding of monotonic accelerated methods.