Optimization with orthogonality constraints problems appear widely in applications from science and engineering. We address these types of problems from an numerical approach. Our new framework combines the steepest gradient descent using implicit information with and operator projection 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 results of our algorithm are proved. The effectiveness of our proposed algorithm is demonstrated on a variety of problems including Rayleigh quotient maximization, heterogeneous quadratics minimization and total energy minimization. Numerical results show that the new procedure can outperform some state–of–the–art solvers on many practically problems.
Report 1, Centro de Investigación en Matemáticas CIMAT A.C., (2020).