The symmetric ADMM with indefinite proximal regularization and its application

Hongchun Sun,Min Sun,Maoying Tian

doi:10.1186/s13660-017-1447-3

Hongchun Sun, Min Sun + Show 1 more

Open Access

https://doi.org/10.1186/s13660-017-1447-3

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Abstract

Due to updating the Lagrangian multiplier twice at each iteration, the symmetric alternating direction method of multipliers (S-ADMM) often performs better than other ADMM-type methods. In practical applications, some proximal terms with positive definite proximal matrices are often added to its subproblems, and it is commonly known that large proximal parameter of the proximal term often results in ‘too-small-step-size’ phenomenon. In this paper, we generalize the proximal matrix from positive definite to indefinite, and propose a new S-ADMM with indefinite proximal regularization (termed IPS-ADMM) for the two-block separable convex programming with linear constraints. Without any additional assumptions, we prove the global convergence of the IPS-ADMM and analyze its worst-case mathcal{O}(1/t) convergence rate in an ergodic sense by the iteration complexity. Finally, some numerical results are included to illustrate the efficiency of the IPS-ADMM.

Highlights

Let Rni stand for an ni-dimensional Euclidean space, and let Xi ⊆ Rni be nonempty, closed and convex set, where i =
We show that for any (r, s) ∈ D, the global convergence of the symmetric alternating direction method of multipliers (S-ADMM) with some indefinite proximal regularization can be guaranteed
In Section, we list the iterative scheme of the IPS-ADMM and prove its convergence results, including the global convergence and the convergence rate

Summary

Introduction

(i = , ) of these subproblems, and as a result we have the following proximal S-ADMM (termed PS-ADMM) For the S-ADMM with r ∈ (– , ), s = , Gao et al [ ] have proved that the feasible set of τ can be expanded to {τ |τ > (r – r + )β A A /(r – r + )}. In Section , we list the iterative scheme of the IPS-ADMM and prove its convergence results, including the global convergence and the convergence rate. We define an auxiliary matrix as follows: G = α τIn – βA A , ( )

Results

Conclusion