Designing Unimodular Codes Via Quadratic Optimization

The NP-hard problem of optimizing a quadratic form over the unimodular vector set arises in radar code design scenarios as well as other active sensing and communication applications. To tackle this problem (which we call unimodular quadratic program (UQP)), several computational approaches are devised and studied. Power method-like iterations are introduced for local optimization of UQP. Furthermore, a monotonically error-bound improving technique (MERIT) is proposed to obtain the global optimum or a local optimum of UQP with good sub-optimality guarantees. The provided sub-optimality guarantees are case-dependent and may outperform the π/4 approximation guarantee of semi-definite relaxation. Several numerical examples are presented to illustrate the performance of the proposed method. The examples show that for several cases, including rank-deficient matrices, the proposed methods can solve UQPs efficiently in the sense of sub-optimality guarantee and computational time.