Abstract:

Optimization problems with orthogonality constraints have a wide range of applications in the field of materials science, statistics and data science. Many optimization algorithms on manifold can be applied to this type of problems, since the feasible region of orthogonal constraint is known as Stiefel manifold.
In recent years, with the expansion of variable scale required by practical application, the limitations of existing methods on manifold are reflected in practice. On the other hand, some efficient approaches based on new concepts are proposed recently. In this paper, we briefly introduce the main classes of methods for optimization problems with orthogonality constraints including  retraction based method, non-retraction based method and infeasible method respectively. We also discuss the main characteristics of these approaches, the scenarios in which these approaches are suitable and the possible directions for further development.
 

Key words: orthogonality constraints, Stiefel manifold, retraction based method