Optimization under Unitary Matrix Constraint using Approximate Matrix Exponential

In many engineering applications we deal with constrained optimization problems w.r.t complex valued matrices. This paper proposes a Riemannian geometry approach for optimization of a real valued cost function J of complex valued matrix argument W, under the constraint that W is an n times n unitary matrix. An approximate steepest descent algorithm based on Taylor series expansion is developed. The approximation satisfies the unitary matrix constraint accurately even if low order approximation is used. Armijo adaptive step size rule (E. Polak, 1997) is used while moving towards the optimum. In the simulation examples, the proposed algorithm is applied to array signal processing and communications problems. The method outperforms other widely used algorithms