  This paper considers the problem of minimizing a convex expectation function
with a set of inequality convex expectation constraints. We present a
computable stochastic approximation type algorithm, namely the stochastic
linearized proximal method of multipliers, to solve this convex stochastic
optimization problem. This algorithm can be roughly viewed as a hybrid of
stochastic approximation and the traditional proximal method of multipliers.
Under mild conditions, we show that this algorithm exhibits $O(K^{-1/2})$
expected convergence rates for both objective reduction and constraint
violation if parameters in the algorithm are properly chosen, where $K$ denotes
the number of iterations. Moreover, we show that, with high probability, the
algorithm has $O(\log(K)K^{-1/2})$ constraint violation bound and
$O(\log^{3/2}(K)K^{-1/2})$ objective bound. Some preliminary numerical results
demonstrate the performance of the proposed algorithm.
