  This paper studies a proximal alternating direction method of multipliers
(ADMM) with variable metric indefinite proximal terms for linearly constrained
convex optimization problems. The proximal ADMM plays an important role in many
application areas, since the subproblems of the method are easy to solve.
Recently, it is reported that the proximal ADMM with a certain fixed indefinite
proximal term is faster than that with a positive semidefinite term, and still
has the global convergence property. On the other hand, Gu and Yamashita
studied a variable metric semidefinite proximal ADMM whose proximal term is
generated by the BFGS update. They reported that a slightly indefinite matrix
also makes the algorithm work well in their numerical experiments. Motivated by
this fact, we consider a variable metric indefinite proximal ADMM, and give
sufficient conditions on the proximal terms for the global convergence.
Moreover, we propose a new indefinite proximal term based on the BFGS update
which can satisfy the conditions for the global convergence.
