A Scaled Gradient Projection Method for Minimization over the Stiefel Manifold

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 eigen-value 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.