Some optimization problems in multivariate statistics

Interesting and important multivariate statistical problems containing principal component analysis, statistical visualization and singular value decomposition, furthermore, one of the basic theorems of linear algebra, the matrix spectral theorem, the characterization of the structural stability of dynamical systems and many others lead to a new class of global optimization problems where the question is to find optimal orthogonal matrices. A special class is where the problem consists in finding, for any 2kn, the dominant k-dimensional eigenspace of an n×n symmetric matrix A in Rⁿ where the eigenspaces are spanned by the k largest eigenvectors. This leads to the maximization of a special quadratic function on the Stiefel manifold M_n,k. Based on the global Lagrange multiplier rule developed in Rapcsák (1997) and the paper dealing with Stiefel manifolds in optimization theory (Rapcsák, 2002), the global optimality conditions of this smooth optimization problem are obtained, then they are applied in concrete cases.