Inertial accelerated augmented Lagrangian algorithms with scaling coefficients to solve exactly and inexactly linearly constrained convex optimization problems

Abstract

In this paper, we propose an inertial accelerated augmented Lagrangian algorithm with a scaling coefficient tailored for solving linearly constrained convex optimization problems. Under suitable scaling conditions, we show that the iteration sequence generated by the algorithm converges to a saddle point of the Lagrangian function. Moreover, we prove fast convergence rates of the Lagrangian residual, the objective residual and the feasibility violation. In the case where the scaling coefficient grows exponentially, we show that the algorithm can achieve linear convergence rates without requiring assumptions of strong convexity or metric subregularity. Additionally, we consider an inexact variant of the proposed algorithm, wherein we find an $ϵ$-stationary solution of the subproblem. Under an additional assumption regarding the error sequence and the scaling coefficient, we prove the preservation of these convergence properties for the inexact variant. Finally, we present some numerical experiments to validate the effectiveness of the proposed algorithms.