Optimization algorithms exploiting unitary constraints

This paper presents novel algorithms that iteratively converge to a local minimum of a real-valued function ( ) subject to the constraint that the columns of the complex-valued matrix are mutually orthogonal and have unit norm. The algorithms are derived by reformulating the constrained optimization problem as an unconstrained one on a suitable manifold. This significantly reduces the dimensionality of the optimization problem. Pertinent features of the proposed framework are illustrated by using the framework to derive an algorithm for computing the eigenvector associated with either the largest or the smallest eigenvalue of a Hermitian matrix.