Nonmonotone feasible arc search algorithm for minimization on Stiefel manifold

We devise a new numerical method for solving the minimization problem over the Stiefel manifold, that is, the set of matrices of order n × p (here p ≤ n ) with orthonormal columns. Our approach consists in a nonmonotone feasible arc search along a sufficient descent direction to assure convergence to stationary points, regardless the initial point considered. The feasibility of the iterates is maintained through a variation of the Cayley transform and thus our scheme can be seen as a retraction-based algorithm for minimization with orthogonality constraints. We emphasize that our scheme solves a p× p linear system at each iteration and has computational complexity of O(np^2) + O(p^3) , which is interesting when p ≪ n . We present a general algorithmic framework for minimization on Stiefel manifold, give its global convergence properties and report numerical experiments on interesting applications.