  In this paper, we aim to solve high dimensional convex quadratic programming
(QP) problems with a large number of quadratic terms, linear equality and
inequality constraints. In order to solve the targeted problems to a desired
accuracy efficiently, we develop a two-phase proximal augmented Lagrangian
method, with Phase I to generate a reasonably good initial point to warm start
Phase II to obtain an accurate solution efficiently. More specifically, in
Phase I, based on the recently developed symmetric Gauss-Seidel (sGS)
decomposition technique, we design a novel sGS based semi-proximal augmented
Lagrangian method for the purpose of finding a solution of low to medium
accuracy. Then, in Phase II, a proximal augmented Lagrangian algorithm is
proposed to obtain a more accurate solution efficiently. Extensive numerical
results evaluating the performance of our proposed algorithm against the highly
optimized commercial solver Gurobi and the open source solver OSQP are
presented to demonstrate the high efficiency and robustness of our proposed
algorithm for solving various classes of large-scale convex QP problems.
