  Despite the numerous uses of semidefinite programming (SDP) and its universal
solvability via interior point methods (IPMs), it is rarely applied to
practical large-scale problems. This mainly owes to the computational cost of
IPMs that increases in a bad exponential way with the data size. While
first-order algorithms such as ADMM can alleviate this issue, but the
scalability improvement appears far not enough. In this work, we aim to achieve
extra acceleration for ADMM by appealing to a non-Euclidean metric space, while
maintaining everything in closed-form expressions. The efficiency gain comes
from the extra degrees of freedom of a variable metric compared to a scalar
step-size, which allows us to capture some additional ill-conditioning
structures.
  On the application side, we consider the quadratically constrained quadratic
program (QCQP), which naturally appears in an SDP form after a dualization
procedure. This technique, known as semidefinite relaxation, has important uses
across different fields, particularly in wireless communications. Numerically,
we observe that the scalability property is significantly improved. Depending
on the data generation process, the extra acceleration can easily surpass the
scalar-parameter efficiency limit, and the advantage is rapidly increasing as
the data conditioning becomes worse.
