A two‐phase strategy for control constrained elliptic optimal control problems

• Published in: Numerical linear algebra with applications Vol. 25; no. 4
• Main Authors: Song, Xiao‐Liang, Yu, Bo
• Format: Journal Article
• Language: English
• Published: Oxford Wiley Subscription Services, Inc 01.08.2018
• Subjects: Algorithms; Control theory; Discretization; Finite element method; inexact semiproximal ADMM; Iterative methods; Optimal control; semismooth Newton method
• ISSN: 1070-5325; 1099-1506
• DOI: 10.1002/nla.2138

Summary Elliptic optimal control problems with pointwise box constraints on the control are considered. To numerically solve elliptic optimal control problems with pointwise box constraints on the control, an inexact alternating direction method of multipliers (iADMM) is first proposed on the continuous level with the aim of solving discretized problems with moderate accuracy. Then, the standard piecewise linear finite element is employed to discretize the related subproblems appearing in each iteration of the iADMM algorithm. Such approach will give us the freedom to discretize two inner subproblems of the iADMM algorithm by different discretized scheme, respectively. More importantly, it should be emphasized that the discretized version of the iADMM algorithm can be regarded as a modification of the inexact semiproximal ADMM (isPADMM) algorithm. In order to obtain more accurate solution, the primal‐dual active set method is used as a postprocessor of the isPADMM. Numerical results not only show that the isPADMM and the two‐phase strategy are highly efficient but also show the mesh independence of the isPADMM.