Iteration Complexity of a Proximal Augmented Lagrangian Method for Solving Nonconvex Composite Optimization Problems with Nonlinear Convex Constraints

Abstract

This paper proposes and analyzes a proximal augmented Lagrangian (NL-IAPIAL) method for solving constrained nonconvex composite optimization problems, where the constraints are smooth and convex with respect to the order given by a closed convex cone. Each NL-IAPIAL iteration consists of inexactly solving a proximal augmented Lagrangian subproblem by an accelerated composite gradient method followed by a Lagrange multiplier update. Under some mild assumptions, a complexity bound for NL-IAPIAL to obtain an approximate stationary solution of the problem is also derived. Numerical experiments are also given to illustrate the computational efficiency of the proposed method. Funding: This work was supported by the Natural Sciences and Engineering Research Council of Canada [Grant PGSD3-516700-2018]; Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grant 312559/2019-4]; Fundação de Amparo à Pesquisa do Estado de Goiás; Office of Naval Research [Grant N00014-18-1-2077]; Exascale Computing Project [Grant 17-SC-20-SC]; Air Force Office of Scientific Research [Grant FA9550-22-1-0088]; UT-Battelle, LLC [Grant DE-AC05-00OR22725].