Convergence Guarantees for First-Order Methods via Lyapunov Analysis in Composite Strong-Weak Convex Optimization

Abstract

Nesterov’s accelerated gradient (NAG) method extends the classical gradient descent algorithm by improving the convergence rate from O(1/t ) to O(1/t2 ) in convex optimization. In this work, we study the proximal gradient framework for additively separable composite objectives consisting of smooth and non-smooth terms. We show that the Nesterov accelerated proximal gradient method (NAPGα) achieves a convergence rate of o(1/t2 ) for strong–weak convex functions when α > 3. A Lyapunov-based analysis is developed to establish the fast convergence of the composite gradient operator in the setting where the smooth component is strongly convex and the non-smooth component is weakly convex. Furthermore, we prove the equivalence between the Nesterov accelerated proximal gradient method and the Ravine accelerated proximal gradient scheme.