An Augmented Lagrangian Method For Nonconvex Optimization Problems with Conically-Convex Constraints

This talk presents and analyzes a novel nonlinear inner accelerated inexact proximal augmented Lagrangian (NL-IAIPAL) method for solving smooth nonconvex composite optimization problems with nonlinear 𝒦-convex constraints, i.e., the constraints are convex with respect to the order given by a closed convex cone 𝒦. Each NL-IAIPAL iteration consists of inexactly solving a proximal augmented Lagrangian subproblem by an accelerated composite gradient (ACG) method followed by a full Lagrange multiplier update. Under some mild assumptions, it is shown that NL-IAIPAL generates an approximate stationary solution of the constrained problem in O(log(1/𝜌)/𝜌³) ACG iterations, where 𝜌 > 0 is a given tolerance. Numerical experiments are given to illustrate the computational efficiency of the presented method.