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In this paper, constrained stochastic optimization problems are considered for the case where the constraint functions are convex (but the criterion function can be non-convex) and the criterion and constraint functions are available only through their noisy observations. A general algorithm of the two time-scale stochastic approximation type is proposed for these problems. The proposed algorithm is applied to Markov decision problems with average cost, average constraints and parameterized stationary policy. The asymptotic behavior of the proposed algorithm is analyzed for the case where the algorithm step-sizes are constant and the noise in the observations of the criterion and constraint functions depends on the algorithm iterates.