A Stochastic Augmented Lagrangian Method for Stochastic Convex Programming (Journal of Computational Mathematics, 2024)
In this paper, we analyze the convergence properties of a stochastic augmented Lagrangian method for solving stochastic convex programming problems with inequalityconstraints. Approximation models for stochastic convex programming problems are constructed from stochastic observations of real objective and constraint functions. Based onrelations between solutions of the primal problem and solutions of the dual problem, itis proved that the convergence of the algorithm from the perspective of the dual problem. Without assumptions on how these random models are generated, when estimatesare merely sufficiently accurate to the real objective and constraint functions with highenough, but fixed, probability, the method converges globally to the optimal solution almost surely. In addition, sufficiently accurate random models are given under differentnoise assumptions. We also report numerical results that show the good performance ofthe algorithm for different convex programming problems with several random models.
Keywords: Stochastic convex optimization; Stochastic approximation; Augmented Lagrangian method; Duality theory.