Title: Polynomial-Time Methods to Solve Unimodular Quadratic Programs With Performance Guarantees,
Abstract: 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.