Symmetric Alternating Direction Method with Indefinite Proximal Regularization for Linearly Constrained Convex Optimization

Abstract

The proximal alternating direction method of multipliers is a popular and useful method for linearly constrained, separable convex problems, especially for the linearized case. In the literature, convergence of the proximal alternating direction method has been established under the assumption that the proximal regularization matrix is positive semi-definite. Recently, it was shown that the regularizing proximal term in the proximal alternating direction method of multipliers does not necessarily have to be positive semi-definite, without any additional assumptions. However, it remains unknown as to whether the indefinite setting is valid for the proximal version of the symmetric alternating direction method of multipliers. In this paper, we confirm that the symmetric alternating direction method of multipliers can also be regularized with an indefinite proximal term. We theoretically prove the global convergence of the indefinite method and establish its worst-case convergence rate in an ergodic sense. In addition, the generalized alternating direction method of multipliers proposed by Eckstein and Bertsekas is a special case in our discussion. Finally, we demonstrate the performance improvements achieved when using the indefinite proximal term through experimental results.