Augmented Lagrangian–Based First-Order Methods for Convex-Constrained Programs with Weakly Convex Objective

First-order methods (FOMs) have been widely used for solving large-scale problems. A majority of existing works focus on problems without constraint or with simple constraints. Several recent studied FOMs complicated functional In this paper, we design a novel augmented Lagrangian (AL)–based FOM nonconvex objective and convex functions. The new method follows the framework proximal point (PP) method. On approximately PP subproblems, it mixes usage inexact AL (iALM) quadratic penalty method, whereas latter is always fed estimated multipliers by iALM. proposed achieves best-known complexity result to produce near Karush–Kuhn–Tucker (KKT) point. Theoretically, hybrid has lower iteration-complexity requirement than its counterpart that only uses iALM solve subproblems; numerically, can perform significantly better pure-penalty-based Numerical experiments are conducted linearly constrained programs. numerical results demonstrate efficiency over ones.