  Certain optimization problems in communication systems, such as
limited-feedback constant-envelope beamforming or noncoherent $M$-ary
phase-shift keying ($M$PSK) sequence detection, result in the maximization of a
fixed-rank positive semidefinite quadratic form over the $M$PSK alphabet. This
form is a special case of the Rayleigh quotient of a matrix and, in general,
its maximization by an $M$PSK sequence is $\mathcal{NP}$-hard. However, if the
rank of the matrix is not a function of its size, then the optimal solution can
be computed with polynomial complexity in the matrix size. In this work, we
develop a new technique to efficiently solve this problem by utilizing
auxiliary continuous-valued angles and partitioning the resulting continuous
space of solutions into a polynomial-size set of regions, each of which
corresponds to a distinct $M$PSK sequence. The sequence that maximizes the
Rayleigh quotient is shown to belong to this polynomial-size set of sequences,
thus efficiently reducing the size of the feasible set from exponential to
polynomial. Based on this analysis, we also develop an algorithm that
constructs this set in polynomial time and show that it is fully
parallelizable, memory efficient, and rank scalable. The proposed algorithm
compares favorably with other solvers for this problem that have appeared
recently in the literature.
