In this talk, we consider the problem of minimizing a continuously differentiable function on the Stiefel manifold. This kind of problem is widely applicable in many fields such as nearest low-rank correlation matrix problem, linear eigenvalue problem, sparse principal component analysis, Kohn-Sham total energy minimization, low-rank matrix completion, image segmentation, dimension reduction techniques in pattern recognition, among others. In order to address this problem, we introduce two adaptive scaled gradient projection methods that incorporate scaling matrices that depend on the step size and a parameter that controls the search direction. These iterative algorithms use a projection operator based on the QR factorization to preserve the feasibility in each iteration. In addition, we consider a Barzilai and Borwein-like step-size combined with the Armijo line-search technique in order to accelerate the convergence of the proposed procedures. We establish the global convergence for these schemes, and we evaluate their effectiveness and efficiency through an extensive computational study, comparing our approaches with other state-of-the-art gradient-type algorithms.

*Text informed by the author.