Polynomial-Time Methods to Solve Unimodular Quadratic Programs With Performance Guarantees

We develop polynomial-time heuristic methods to solve unimodular quadratic programs (UQPs) approximately, which are known to be NP-hard. In the UQP framework, we maximize a quadratic function of a vector of complex variables with unit modulus. Several problems in active sensing and wireless communication applications boil down to UQP. With this motivation, we present three new heuristic methods with polynomial-time complexity to solve the UQP approximately. The first method is called dominant-eigenvector-matching; here the solution is picked that matches the complex arguments of the dominant eigenvector of the Hermitian matrix in the UQP formulation. We also provide a performance guarantee for this method. The second method, a greedy strategy, is shown to provide a performance guarantee of (1-1/e) with respect to the optimal objective value given that the objective function possesses a property called string submodularity. The third heuristic method is called row-swap greedy strategy, which is an extension to the greedy strategy and utilizes certain properties of the UQP to provide a better performance than the greedy strategy at the expense of an increase in computational complexity. We present numerical results to demonstrate the performance of these heuristic methods, and also compare the performance of these methods against a standard heuristic method called semidefinite relaxation

RIS-Assisted MISO Communication: Optimal Beamformers and Performance Analysis

We study a multiple-input single-output (MISO) communication system assisted by a reconfigurable intelligent surface (RIS). A base station (BS) having multiple antennas is assumed to be communicating to a single-antenna user equipment (UE), with the help of a RIS. We assume that the system operates in an environment with line-of-sight (LoS) between the BS and RIS, whereas the RIS-UE link experiences Rayleigh fading. We present a closed form expression for the optimal active and passive beamforming vectors at the BS and RIS respectively. Then, by characterizing the statistical properties of the received SNR at the UE, we apply them to derive analytical approximations for different system performance measures, including the outage probability, average achievable rate and average symbol error probability (SEP). Our results, in general, demonstrate that the gain due to RIS can be substantial, and can be significantly greater than the gains reaped by using multiple BS antennas.Comment: Accepted for publication in IEEE GlobeCom-202