A Model-Based Stochastic Augmented Lagrangian Method for Online Stochastic Optimization

In this paper, we focus on online stochastic optimization problems in which random parameters follow time-varying distributions. In each round t, a decision is obtained from solving the current optimization problem. Then samples are drawn from distributions which are updated after obtaining the decision. The objective and constraint are updated in this process, and the updated problem is used to obtain the next decision. To solve the online stochastic optimization problem, we propose a model-based stochastic augmented Lagrangian method, which is referred to as the MSALM. In each round, we construct model functions for the sample objective and constraint functions based on their properties, which reduce computational complexity. The step size is designed in a dynamic way and decreases as t increases to accelerate convergence. Due to the setting of the online stochastic problem, we use stochastic dynamic regret and constraint violation to measure the performance of our algorithm. Under certain assumptions, we prove that our algorithm’s stochastic dynamic regret and constraint violation have a sublinear bound in terms of the total number of slots T. We design simulation experiments to verify the efficiency of our online algorithm. Its performance is evaluated on a range of information and system engineering problems, including adaptive filtering, online logistic regression, time-varying smart grid energy dispatch, online network resource allocation, and path planning. In addition, in the context of the path planning problem, we integrate our algorithm with supervised learning to demonstrate its enhanced capabilities. The experimental results validate the performance of our new algorithm in practical applications.