Fixed-rank Rayleigh Quotient Maximization by an MMPSK Sequence

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.Comment: 15 pages, 12 figures, To appear in IEEE Transactions on Communication

Rank-Deficient Quadratic-Form Maximization Over M-Phase Alphabet: Polynomial-Complexity Solvability And Algorithmic Developments

The maximization of a positive (semi) definite complex quadratic form over a finite alphabet is NP-hard and achieved through exhaustive search when the form has full rank. However, if the form is rank-deficient, the optimal solution can be computed with only polynomial complexity in the length N of the maximizing vector. In this work, we consider the general case of a rank-D positive (semi) definite complex quadratic form and develop a method that maximizes the form with respect to a M-phase vector with polynomial complexity. The proposed method efficiently reduces the size of the feasible set from exponential to polynomial. We also develop an algorithm that constructs the polynomial-size candidate set in polynomial time and observe that it is fully parallelizable and rank-scalable