Faster augmented Lagrangian method for solving convex optimization problems with linear constraints

Abstract

A faster augmented Lagrangian method (Faster ALM) with a relaxed parameter ranging from 0 to 2 is introduced in this paper for solving convex optimization problems with equality constraints. The proposed Faster ALM demonstrates a convergence rate of $O\left(1/\sum_{i=0}^{k}a_i\right)$, where ($a_i>0$ is an arbitrary constant), in a non-ergodic sense of the Lagrangian primal-dual gap, the objective function value, and the feasibility measure.To further reduce the computational cost in each iteration, we present the Linearized Faster ALM, which involves linearizing the augmented Lagrangian term while maintaining a relaxed parameter within the range of 0 to 2. The proposed Linearized Faster ALM exhibits a convergence rate of $O(1/k)$ in a non-ergodic sense of the Lagrangian primal-dual gap, the objective function value, and the feasibility measure. MSC Classification: 47H09 , 47H10 , 90C25 , 90C30