An inexact variable metric proximal linearization method for composite optimization on manifolds

Abstract

This paper concerns the minimization of the composition of a nonsmooth convex function and a $\mathcal{C}^{1,1}$ mapping $F$ over a $\mathcal{C}^2$-smooth embedded closed submanifold $\mathcal{M}$. For this class of nonconvex and nonsmooth problems, we propose an inexact variable metric proximal linearization method by leveraging its composite structure and the retraction and first-order information of $\mathcal{M}$, which at each iteration seeks an inexact solution to a subspace constrained strongly convex problem by a practical inexactness criterion. Under the boundedness assumption on the iterate sequence, we establish the $O(\epsilon^{-3})$ oracle complexity with a dual fast gradient method as the inner solver, and prove that any cluster point of the iterate sequence is a stationary point. If in addition the constructed potential function has the Kurdyka-Lojasiewicz (KL) property on the set of cluster points, the iterate sequence converges to a stationary point, and if the potential function has the KL property of exponent $q\in[\frac{1}{2},1)$, the local convergence rate is characterized. We also provide a condition only involving the original data to identify the KL property of the potential function with an exponent $q\in[0,1)$. Numerical comparisons with the existing methods validate the efficiency of the proposed method.