Designing Unimodular Codes via Quadratic Optimization is not Always Hard

TL;DR

This paper introduces new optimization techniques for the NP-hard problem of unimodular quadratic programming, demonstrating improved solutions and guarantees over existing methods in radar code design applications.

Contribution

It proposes a specialized local optimization scheme and the MERIT method, offering better sub-optimality guarantees and efficiency for solving UQP.

Findings

MERIT can efficiently solve UQP with good sub-optimality guarantees.

The proposed methods outperform semi-definite relaxation in case-dependent scenarios.

Numerical examples confirm the effectiveness of the approaches in radar code design contexts.

Abstract

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 programming (UQP)), several computational approaches are devised and studied. A specialized local optimization scheme for UQP is introduced and shown to yield superior results compared to general local optimization methods. Furthermore, a \textbf{m}onotonically \textbf{er}ror-bound \textbf{i}mproving \textbf{t}echnique (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 generally outperform the $\pi/4$ approximation guarantee of semi-definite relaxation. Several numerical examples are presented to illustrate the performance of the proposed method. The examples show that for cases including several matrix structures used in radar code design, MERIT can solve UQP efficiently in the sense of sub-optimality guarantee and computational time.