Iteration-complexity of a proximal augmented Lagrangian method for solving nonconvex composite optimization problems with nonlinear convex constraints

TL;DR

This paper introduces a proximal augmented Lagrangian method for nonconvex composite optimization with nonlinear convex constraints, providing iteration complexity analysis and demonstrating efficiency through numerical experiments.

Contribution

It develops a novel NL-IAPIAL method with proven iteration complexity for solving nonconvex problems with nonlinear convex constraints.

Findings

Achieves approximate stationary solutions within specified tolerance.

Iteration complexity is ${\

Numerical experiments confirm computational efficiency.

Abstract

This paper proposes and analyzes a proximal augmented Lagrangian (NL-IAPIAL) method for solving smooth nonconvex composite optimization problems with nonlinear $\cal K$-convex constraints, i.e., the constraints are convex with respect to the order given by a closed convex cone $\cal K$. Each NL-IAPIAL iteration consists of inexactly solving a proximal augmented Lagrangian subproblem by an accelerated composite gradient (ACG) method followed by a Lagrange multiplier update. Under some mild assumptions, it is shown that NL-IAPIAL generates an approximate stationary solution of the constrained problem in ${\cal O}(\log(1/\rho)/\rho^{3})$ inner iterations, where $\rho>0$ is a given tolerance. Numerical experiments are also given to illustrate the computational efficiency of the proposed method.