On minimization on Stiefel manifolds

The minimization of a smooth function f : R-kn --> R under the constraint that vectors x(1), x(2),....,x(k) is an element of R-n, k less than or equal to n, form an orthonormal system seems to be a new and interesting global optimization problem with important theoretical and practical applications. The set of feasible points determines a differentiable manifold introduced by Stiefel in 1935. Based on the nice geometric structure. the optimality conditions are obtained by the global Lagrange multiplier rule, and global optimality conditions based on local information. which make the advantages of using the Riemannian geometry in difficult smooth optimization problems clear. (C) 2002 Elsevier Science B.V. All rights reserved.