A Multilevel Heterogeneous ADMM Algorithm for Elliptic Optimal Control Problems with L 1 -Control Cost

Xiaotong Chen, Xiaoliang Song, Zixuan Chen () and Lijun Xu
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Xiaotong Chen: School of Science, Dalian Maritime University, Dalian 116026, China
Xiaoliang Song: School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
Zixuan Chen: College of Sciences, Northeastern University, Shenyang 110819, China
Lijun Xu: School of Science, Dalian Maritime University, Dalian 116026, China

Mathematics, 2023, vol. 11, issue 3, 1-21

Abstract: In this paper, elliptic optimal control problems with L 1 -control cost and box constraints on the control are considered. To numerically solve the optimal control problems, we use the First optimize, then discretize approach. We focus on the inexact alternating direction method of multipliers (iADMM) and employ the standard piecewise linear finite element approach to discretize the subproblems in each iteration. However, in general, solving the subproblems is expensive, especially when the discretization is at a fine level. Motivated by the efficiency of the multigrid method for solving large-scale problems, we combine the multigrid strategy with the iADMM algorithm. Instead of fixing the mesh size before the computation process, we propose the strategy of gradually refining the grid. Moreover, to overcome the difficulty whereby the L 1 -norm does not have a decoupled form, we apply nodal quadrature formulas to approximately discretize the L 1 -norm and L 2 -norm. Based on these strategies, an efficient multilevel heterogeneous ADMM (mhADMM) algorithm is proposed. The total error of the mhADMM consists of two parts: the discretization error resulting from the finite-element discretization and the iteration error resulting from solving the discretized subproblems. Both errors can be regarded as the error of inexactly solving infinite-dimensional subproblems. Thus, the mhADMM can be regarded as the iADMM in function space. Furthermore, theoretical results on the global convergence, as well as the iteration complexity results o ( 1 / k ) for the mhADMM, are given. Numerical results show the efficiency of the mhADMM algorithm.

Keywords: optimal control problems; ADMM; sparse regularization; multilevel (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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