A variable smoothing for weakly convex composite minimization with manifold constraint via parametrization

Abstract In this paper, we address a manifold constrained nonsmooth optimization problem involving the composition of a weakly convex function and a smooth mapping under the availability of a parametrization of the manifold. To find a stationary point of the target problem, we propose a variable smoothing-type algorithm by combining the ideas of (i) translating the constrained problem into a Euclidean optimization problem with a parametrization of the constraint set; (ii) exploiting a sequence of smooth approximations of the objective, constructed using the Moreau envelope. The proposed algorithm produces a vector sequence by the gradient descent update of a smooth approximation of the objective. In a case where the proximity operator of the weakly convex function is available, the proposed algorithm does not require any iterative solver for subproblems therein. By leveraging tools in the variational analysis, we show the so-called gradient consistency property , which is a key ingredient for smoothing-type algorithms, of the smooth approximations used in this paper. Based on the gradient consistency property, we establish a subsequential convergence guarantee of the proposed algorithm to a stationary point of the parameterized problem and clarify conditions, such as a submersion condition on the parametrization, under which the corresponding point is stationary for the original manifold constrained problem. Numerical experiments demonstrate the efficacy of the proposed algorithm.