Jie Liu, Jiawei Chen, Jinlan Zheng, Xuerui Zhang, Zhongping Wan, A new accelerated positive-indefinite proximal ADMM for constrained separable convex optimization problems

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DOI: 10.23952/jnva.6.2022.6.08

Volume 6, Issue 6, 1 December 2022, Pages 707-723

Abstract. The alternating direction method of multipliers (ADMM) is a powerful method to solve constrained convex optimization problems with the separable structure. The ADMM with the positive-indefinite proximal terms, which has ergodic convergent rate $O(\frac{1}{K})$ with the number of iterations $K$, is more general than the ADMM with positive-definite proximal terms. In this paper, we propose a new accelerated positive-indefinite proximal linearized ADMM algorithm with positive-indefinite proximal matrix by the techniques of extrapolation. We obtain the nonergodic convergence rate $O(\frac{1}{K})$ in the sense of objective values and the nonergodic convergence rate $O(\frac{1}{\sqrt{K}})$ in the sense of iterative sequence of the proposed method as well as the upper bound of the violation of constraints. Numerical results are reported to show the efficiency of the proposed method.