Implicit steepest descent algorithm for optimization with orthogonality constraints

Optimization problems with orthogonality constraints appear widely in applications from science and engineering. We address these types of problems from a numerical approach. Our new framework combines the steepest gradient descent, using implicit information, with a projection operator in order to construct a feasible sequence of points. In addition, we adopt an adaptive Barzilai–Borwein steplength mixed with a globalization technique in order to speed-up the convergence of our procedure. The global convergence, and some theoretical related to our algorithm are proved. The effectiveness of our proposed algorithm is demonstrated on a variety of problems including Rayleigh quotient maximization, heterogeneous quadratics minimization, weighted orthogonal procrustes problems and total energy minimization. Numerical results show that the new procedure can outperform some state of the art solvers on some practically problems.