First-Order Methods for Constrained Convex Programming Based on Linearized Augmented Lagrangian Function

First-order methods (FOMs) have been popularly used for solving large-scale problems. However, many existing works only consider unconstrained problems or those with simple constraint. In this paper, we develop two FOMs constrained convex programs, where the constraint set is represented by affine equations and smooth nonlinear inequalities. Both are based on classical augmented Lagrangian function. They update multipliers in same way as method (ALM) but use different primal updates. The first method, at each iteration, performs a single proximal gradient step to variable, second block version of one. For establish its global iterate convergence sublinear local linear convergence, show result expectation. Numerical experiments carried out basis pursuit denoising, quadratically quadratic Neyman-Pearson classification problem empirical performance proposed methods. Their numerical behaviors closely match established theoretical results.