  The augmented Lagrangian method (ALM) is a benchmark for convex programming
problems with linear constraints; ALM and its variants for linearly
equality-constrained convex minimization models have been well studied in the
literature. However, much less attention has been paid to ALM for efficiently
solving linearly inequality-constrained convex minimization models. In this
paper, we exploit an enlightening reformulation of the newly developed
indefinite linearized ALM for the equality-constrained convex optimization
problem, and present a new indefinite linearized ALM scheme for efficiently
solving the convex optimization problem with linear inequality constraints. The
proposed method enjoys great advantages, especially for large-scale
optimization cases, in two folds mainly: first, it largely simplifies the
challenging key subproblem of the classic ALM by employing its linearized
reformulation, while keeping low complexity in computation; second, we show
that only a smaller proximity regularization term is needed for provable
convergence, which allows a bigger step-size and hence significantly better
performance. Moreover, we show the global convergence of the proposed scheme
upon its equivalent compact expression of prediction-correction, along with a
worst-case $\mathcal{O}(1/N)$ convergence rate. Numerical results on some
application problems demonstrate that a smaller regularization term can lead to
a better experimental performance, which further confirms the theoretical
results presented in this study.
