In this paper, Elliptic control problems with pointwise box constraints on the state is considered, where the corresponding Lagrange multipliers in general only represent regular Borel measure functions. To tackle this difficulty, the Lavrentiev regularization is employed to deal with the state constraints. To numerically discretize the resulted problem, full piecewise linear finite element discretization is employed. Estimation of the error produced by regularization and discretization is done. The error order of full discretization is not inferior to that of variational discretization because of the Lavrentiev-regularization. Taking the discretization error into account, algorithms of high precision do not make much sense. Utilizing efficient first-order algorithms to solve discretized problems to moderate accuracy is sufficient. Then a heterogeneous alternating direction method of multipliers (hADMM) is proposed. Different from the classical ADMM, our hADMM adopts two different weighted norms in two subproblems respectively. Additionally, to get more accurate solution, a two-phase strategy is presented, in which the primal-dual active set (PDAS) method is used as a postprocessor of the hADMM. Numerical results not only verify error estimates but also show the efficiency of the hADMM and the two-phase strategy.
Mots-clés : Optimal control, Pointwise state constraints, Lavrentiev regularization, Error estimates, Heterogeneous ADMM, Two-phase strategy
@article{COCV_2019__25__A5_0,
author = {Chen, Zixuan and Song, Xiaoliang and Zhang, Xuping and Yu, Bo},
title = {A {FE-ADMM} algorithm for {Lavrentiev-regularized} state-constrained elliptic control problem},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {25},
year = {2019},
doi = {10.1051/cocv/2018019},
zbl = {1437.49003},
mrnumber = {3943363},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2018019/}
}
TY - JOUR AU - Chen, Zixuan AU - Song, Xiaoliang AU - Zhang, Xuping AU - Yu, Bo TI - A FE-ADMM algorithm for Lavrentiev-regularized state-constrained elliptic control problem JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2018019/ DO - 10.1051/cocv/2018019 LA - en ID - COCV_2019__25__A5_0 ER -
%0 Journal Article %A Chen, Zixuan %A Song, Xiaoliang %A Zhang, Xuping %A Yu, Bo %T A FE-ADMM algorithm for Lavrentiev-regularized state-constrained elliptic control problem %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2018019/ %R 10.1051/cocv/2018019 %G en %F COCV_2019__25__A5_0
Chen, Zixuan; Song, Xiaoliang; Zhang, Xuping; Yu, Bo. A FE-ADMM algorithm for Lavrentiev-regularized state-constrained elliptic control problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 5. doi : 10.1051/cocv/2018019. http://www.numdam.org/articles/10.1051/cocv/2018019/
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