The symmetric ADMM with indefinite proximal regularization and its application

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 [Formula: see text] convergence rate in an ergodic sense by the iteration complexity. Finally, some numerical results are included to illustrate the efficiency of the IPS-ADMM.