Optimization algorithms exploiting unitary constraints

• Published in: IEEE transactions on signal processing Vol. 50; no. 3; pp. 635 - 650
• Main Author: Manton, J.H.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.03.2002; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Constraint optimization; Cost function; Eigenvalues; Eigenvalues and eigenfunctions; Eigenvectors; Exact sciences and technology; Frequency estimation; Information, signal and communications theory; Iterative algorithms; Manifolds; Mathematical analysis; Mathematical methods; Norms; Operational research and scientific management; Operational research. Management science; Optimization; Optimization. Search problems; Signal processing; Signal processing algorithms; Space technology; Subspace constraints; Symmetric matrices; Telecommunications and information theory; Performance evaluation; Simulation; Steepest descent method; Convergence rate; Signal processing; Eigenvalue problem; Algorithm; Newton method; Constrained optimization
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/78.984753

This paper presents novel algorithms that iteratively converge to a local minimum of a real-valued function f (X) subject to the constraint that the columns of the complex-valued matrix X 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.