Q-Linear Convergence of the Proximal Augmented Lagrangian Method for Non-Convex Conic Programming

Abstract: This paper provides a local convergence analysis of the proximal augmented Lagrangian method (PALM) applied to a class of non-convex conic programming problems. Previous convergence results for PALM typically imposed assumptions such as constraint non-degeneracy, strict complementarity, second-order sufficiency conditions, or a combination of constraint nondegeneracy with strong second-order sufficiency conditions. In contrast, our work demonstrates a Q-linear convergence rate for an inexact version of PALM in the context of non-convex conic programming, without requiring the uniqueness of the Lagrange multipliers. The analysis relies solely on the second-order sufficiency condition and the calmness property of the multiplier mapping, presenting a more relaxed set of conditions for ensuring convergence.