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We present theorems and an algorithm to find optimal or near-optimal ldquostochastic resonancerdquo (SR) noise benefits for Neyman-Pearson hypothesis testing and for more general inequality-constrained signal detection problems. The optimal SR noise distribution is just the randomization of two noise realizations when the optimal noise exists for a single inequality constraint on the average cost. The theorems give necessary and sufficient conditions for the existence of such optimal SR noise in inequality-constrained signal detectors. There exists a sequence of noise variables whose detection performance limit is optimal when such noise does not exist. Another theorem gives sufficient conditions for SR noise benefits in Neyman-Pearson and other signal detection problems with inequality cost constraints. An upper bound limits the number of iterations that the algorithm requires to find near-optimal noise. The appendix presents the proofs of the main results.