A Scaled Gradient Projection Method for Minimization over the Stiefel Manifold

Abstract

In this paper we consider a class of iterative gradient projection methods for solving optimization problems with orthogonality constraints. The proposed method can be seen as a forward-backward gradient projection method which is an extension of a gradient method based on the Cayley transform. The proposal incorporates a self-adaptive scaling matrix and the Barzilai-Borwein step-sizes that accelerate the convergence of the method. In order to preserve feasibility, we adopt a projection operator based on the QR factorization. We demonstrate the efficiency of our procedure in several test problems including eigenvalue computations and sparse principal component analysis. Numerical comparisons show that our proposal is effective for solving these kind of problems and presents competitive results compared with some state-of-art methods.