TY - JOUR
T1 - A Stochastic Augmented Lagrangian Method for Stochastic Convex Programming
AU - Wang , Jiani
AU - Zhang , Liwei
JO - Journal of Computational Mathematics
VL - 2
SP - 315
EP - 344
PY - 2024
DA - 2024/11
SN - 43
DO - http://doi.org/10.4208/jcm.2208-m2022-0035
UR - https://global-sci.org/intro/article_detail/jcm/23540.html
KW - Stochastic convex optimization, Stochastic approximation, Augmented Lagrangian method, Duality theory.
AB - <p style="text-align: justify;">In this paper, we analyze the convergence properties of a stochastic augmented Lagrangian method for solving stochastic convex programming problems with inequality
constraints. Approximation models for stochastic convex programming problems are constructed from stochastic observations of real objective and constraint functions. Based on
relations between solutions of the primal problem and solutions of the dual problem, it
is proved that the convergence of the algorithm from the perspective of the dual problem. Without assumptions on how these random models are generated, when estimates
are merely sufficiently accurate to the real objective and constraint functions with high
enough, but fixed, probability, the method converges globally to the optimal solution almost surely. In addition, sufficiently accurate random models are given under different
noise assumptions. We also report numerical results that show the good performance of
the algorithm for different convex programming problems with several random models.</p>