Modified steepest descent and Newton algorithms for orthogonally constrained optimisation. Part I. The complex Stiefel manifold

The classical steepest descent and Newton algorithms can be used to minimise a cost function f(X). This paper shows how they can be modified to take into account the constraint that the columns of the complex-valued matrix X are mutually orthogonal and have unit norm. The algorithms are derived by converting the constrained optimisation problem into an unconstrained one on the Stiefel manifold. This significantly reduces the dimension of the optimisation problem and often results in faster convergence.