PERFORMANCE ANALYSIS OF QUASI INTEGER LEAST SQUARES SOLVERS BASED ON SEMIDEFINITE RELAXATION∗

We consider a random Integer Least Squares (ILS) problem. This NP-hard problem naturally arises in digital communications as the maximum-likelihood detection problem. We analyze two probabilistic quasi-ILS algorithms based on semidefinite relaxations: the SDR algorithm for binary variables and the PSK algorithm for constant modulus variables. Both algorithms are capable of delivering a near-optimal solution with a polynomial worst-case complexity. For a general class of random parameters, we prove that the SDR algorithm provides a constant factor approximation in terms of the objective value, and this constant factor remains bounded with increasing problem size. For the PSK algorithm we show that each local maximum of the low-rank semidefinite relaxation that is feasible for the ILS problem achieves at least a half of the minimum relative objective value, and for the binary case even yields an exact ILS solution. Our analysis shows that the ILS problem can be well approximated in polynomial time by the two semidefinite relaxation strategies.

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