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Orthogonal space-time block codes (OSTBCs) have attracted much attention because they provide an effective and simple scheme for fully utilizing the diversity gain in multi-antenna systems. We address the problem of decoding OSTBCs without channel state information. We place our emphasis on the blind maximum-likelihood (ML) method with the BPSK constellation, and show that blind ML decoding requires the solution of a computationally hard optimization problem. To overcome this computational difficulty, we propose using a high-precision and efficient approximation algorithm, called semidefinite relaxation (SDR), to implement blind ML decoding suboptimally. The resultant SDR-ML blind decoder is efficient in that its complexity is approximately cubic in the number of symbols processed, and is promising for its appealing theoretical worst-case approximation accuracy. Simulation results show that the bit error performance of the SDR-ML blind decoder is substantially better than that of several other blind decoders including the cyclic ML method and the subspace method.