Spectral Projected Gradient Method for the Procrustes Problem

We study and analyze a nonmonotone globally convergent method for minimization on closed sets. This method is based on the ideas from trust-region and Levenberg-Marquardt methods. Thus, the subproblems consists in minimizing a quadratic model of the objective function subject to a given constraint set. We incorporate concepts of bidiagonalization and calculation of the SVD “with inaccuracy” to improve the performance of the algorithm, since the solution of the subproblem by traditional techniques, which is required in each iteration, is computationally expensive. Other feasible methods are mentioned, including a curvilinear search algorithm and a minimization along geodesics algorithm. Finally, we illustrate the numerical performance of the methods when applied to the Orthogonal Procrustes Problem.