A dual-primal balanced augmented Lagrangian method for linearly constrained convex programming

Abstract

Most recently, a balanced augmented Lagrangian method (ALM) has been proposed by He and Yuan for the canonical convex minimization problem with linear constraints, which advances the original ALM by balancing its subproblems, improving its implementation and enlarging its applicable range. In this paper, we propose a dual-primal version of the newly developed balanced ALM, which updates the new iterate via a conversely dual-primal iterative order formally. The new algorithm inherits all advantages of the prototype balanced ALM, and it can be extended to more general separable convex programming problems with both linear equality and inequality constraints. The convergence analysis of the proposed method can be well conducted in the context of variational inequalities. In particular, by some application problems, we numerically validate that these balanced ALM type methods can outperform existing algorithms of the same kind significantly.