A Proximal Augmented Lagrangian Method Based on Quadratic Approximations for Weakly Convex Optimization

Abstract: This paper proposes QPALM, a proximal augmented Lagrangian method based on quadratic approximations, for solving nonlinear programming problems with weakly convex objective and constraint functions. The algorithm is constructed by incorporating quadratic approximations of both the objective and constraint functions into a proximal Lagrangian framework. We establish its non-asymptotic convergence rate in terms of the total number of subproblems solved. The convergence of QPALM is characterized by three metrics associated with the $\varepsilon$-KKT conditions: the squared norm of the gradient of the Moreau envelope of the Lagrangian, the average constraint violation, and the average complementarity violation. All three metrics are shown to converge at a rate of $O(T^{-1/3})$ after $T$ iterations. Preliminary numerical results demonstrate the practical efficiency of the proposed method. These results are established under two mild conditions: (i) weak convexity of all problem functions, and (ii) the existence of a strictly feasible point. The proposed QPALM is a sequentially strongly convex programming method that is readily implementable.