Stochastic Proximal Linearized ADMM for Sparse Distribution Control Problem Constrained by Random Elliptic Equation

Abstract

In this paper, we consider the sparse distributed control problem constrained by a random elliptic equation, which we reformulate as a nonsmooth stochastic optimization problem in Hilbert space. By incorporating the advantages of the stochastic approximation approach and the alternating direction method of multipliers (ADMM), we propose a stochastic ADMM algorithm. This method decouples the stochasticity arising from the random equation constraint from the nonsmoothness of the control objective, allowing them to be tackled separately within the iterations. We introduce stochastic gradients and develop a proximal linearization technique for the stochastic subproblem, allowing each subproblem to admit a closed-form solution. The convergence and a high-probability bound of the proposed method are analyzed for the model problem. Numerical results demonstrate the effectiveness and efficiency of our method.