Modified steepest descent and Newton algorithms for orthogonally constrained optimisation. Part II. The complex Grassmann manifold

Abstract

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. It is assumed that the cost function satisfies f(XQ) = f(X) for any unitary matrix Q. This allows the constrained optimisation problem to be converted into an unconstrained one on the Grassmann manifold. This significantly reduces the dimension of the optimisation problem and often results in faster convergence.