An inexact augmented Lagrangian method for nonsmooth optimization on Riemannian manifold

Abstract: We consider a nonsmooth optimization problem on Riemannian manifold, whose objective function is the sum of a differentiable component and a nonsmooth convex function. We propose a manifold inexact augmented Lagrangian method (MIALM) for the considered problem. The problem is reformulated to a separable form. By utilizing the Moreau envelope, we get a smoothing subproblem at each iteration of the proposed method. Theoretically, under suitable assumptions, the convergence to critical point of the proposed method is established. In particular, under the condition of that the approximate global minimizer of the iteration subproblem could be obtained, we prove the convergence to global minimizer of the origin problem. Numerical experiments show that, the MIALM is a competitive method compared to some existing methods.