An inexact variable metric proximal linearization method for composite optimization over embedded submanifolds

This paper concerns the minimization of the composition of a nonsmooth convex function and a C^1, 1 mapping over a C²-smooth embedded closed submanifold 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 M, which at each iteration seeks an inexact solution to a subspace constrained strongly convex problem by a practical inexactness criterion. Under the common restricted level boundedness assumption, we establish the O (ε^-2) iteration complexity and the O (ε^-2) calls to the subproblem solver for returning an ε-stationary point, and prove that any cluster point of the iterate sequence is a stationary point. If in addition the constructed potential function has the Kurdyka-Łojasiewicz (KL) property on the set of cluster points, the iterate sequence is shown to converge to a stationary point, and if it has the KL property of exponent q1/2, 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 exponent q (0, 1). Numerical comparisons with RiADMM in Li et al. [25 and RiALM in Xu et al. 44 validate the efficiency of the proposed method.