  We consider the convex minimization model with both linear equality and
inequality constraints, and reshape the classic augmented Lagrangian method
(ALM) by balancing its subproblems. As a result, one of its subproblems
decouples the objective function and the coefficient matrix without any extra
condition, and the other subproblem becomes a positive definite system of
linear equations or a positive definite linear complementary problem. The
balanced ALM advances the classic ALM by enlarging its applicable range,
balancing its subproblems, and improving its implementation. We also extend our
discussion to two-block and multiple-block separable convex programming models,
and accordingly design various splitting versions of the balanced ALM for these
separable models. Convergence analysis for the balanced ALM and its splitting
versions is conducted in the context of variational inequalities through the
lens of the classic proximal point algorithm.
