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This paper considers the problem of minimizing a convex expectation function with a set of inequality convex expectation constraints. We propose a stochastic augmented Lagrangian-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 a O (log (K) K − 1 / 2) constraint violation bound and O (log 3 / 2 (K) K − 1 / 2) objective bound. Numerical results demonstrate that the proposed algorithm is efficient. History: Accepted byAntonio Frangioni, Area Editor for Design & Analysis ofAlgorithms—Continuous. Funding: This work was supported by the National Natural Science Foundation of China [Grants 11971089, 11731013, 12071055, and 11871135]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplementary Information [https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.1228] or is available from the IJOC GitHub software repository (https://github.com/INFORMSJoC) at http://dx.doi.org/10.5281/zenodo.6818229. [ABSTRACT FROM AUTHOR]