Abstract
This paper presents the problem of maximizing the determinant of a real K x K-matrix B, subject to the constraint that each row b(k) of B satisfies b(k)(t) Gamma(k)b(k) <= 1, where Gamma(1),..., Gamma(K) are K given real symmetric positive definite matrices. This problem comes from a specific blind signal separation approach, but the criterion differs from approximate diagonalization criteria usually encountered in this area. Furthermore our criterion corresponds to the following nice geometrical problem: given K ellipsoids in R-K, epsilon(k) = {x : x(t)Gamma(k)x <= 1}, k = 1,..., K, find K vectors, b(1) is an element of epsilon(1),..., b(K) is an element of epsilon(K), such that the volume of the parallelepiped defined by these vectors is maximum. Existence and uniqueness of the solution are discussed. An iterative algorithm, based on a relaxation technique, is proposed in order to solve this problem, and its convergence is proved under a simple sufficient condition. Some numerical experiments are performed showing the behavior of the algorithm and its comparison with Newton's methods for nonlinear optimization.