Volume 14, Issue 3
A Symmetric Inertial Alternating Direction Method of Multipliers for Elliptic Equation Constrained Optimization Problem

Adv. Appl. Math. Mech., 14 (2022), pp. 596-621.

Published online: 2022-02

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• Abstract

A new algorithm, called symmetric inertial alternating direction method of multipliers (SIADMM), is designed for separable convex optimization problems with linear constraints in this paper. The convergence rate of the SIADMM is proved to be $\mathcal{O}(1/ \sqrt{k})$. Two kinds of elliptic equation constrained optimization problems, the unconstrained cases as well as the box-constrained cases of the distributed control and the Robin boundary control, are analyzed theoretically and solved numerically. First, the existence and uniqueness of the solutions to these problems are proved. Second, these continuous optimization problems are transformed into discrete optimization problems by the finite element method, and then the discrete optimization problems are solved by the proposed SIADMM. Numerical experiments with different problems are investigated to demonstrate the efficiency of the SIADMM. And the numerical performance of the SIADMM is better than the performance of the ADMM. Moreover, the numerical results show that the convergence rate of the SIADMM tends to be faster than $\mathcal{O}(1/ \sqrt{k})$ in calculation process.

• Keywords

Symmetric inertial alternating direction method of multipliers, convergence rate, elliptic equation constraint, finite element method.

49M37, 65K10, 65K60, 90C25, 90C33

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@Article{AAMM-14-596, author = {Wu , MengyueAi , WenbaoYuan , Jianhua and Tian , Hui}, title = {A Symmetric Inertial Alternating Direction Method of Multipliers for Elliptic Equation Constrained Optimization Problem}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2022}, volume = {14}, number = {3}, pages = {596--621}, abstract = {

A new algorithm, called symmetric inertial alternating direction method of multipliers (SIADMM), is designed for separable convex optimization problems with linear constraints in this paper. The convergence rate of the SIADMM is proved to be $\mathcal{O}(1/ \sqrt{k})$. Two kinds of elliptic equation constrained optimization problems, the unconstrained cases as well as the box-constrained cases of the distributed control and the Robin boundary control, are analyzed theoretically and solved numerically. First, the existence and uniqueness of the solutions to these problems are proved. Second, these continuous optimization problems are transformed into discrete optimization problems by the finite element method, and then the discrete optimization problems are solved by the proposed SIADMM. Numerical experiments with different problems are investigated to demonstrate the efficiency of the SIADMM. And the numerical performance of the SIADMM is better than the performance of the ADMM. Moreover, the numerical results show that the convergence rate of the SIADMM tends to be faster than $\mathcal{O}(1/ \sqrt{k})$ in calculation process.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2020-0400}, url = {http://global-sci.org/intro/article_detail/aamm/20277.html} }
TY - JOUR T1 - A Symmetric Inertial Alternating Direction Method of Multipliers for Elliptic Equation Constrained Optimization Problem AU - Wu , Mengyue AU - Ai , Wenbao AU - Yuan , Jianhua AU - Tian , Hui JO - Advances in Applied Mathematics and Mechanics VL - 3 SP - 596 EP - 621 PY - 2022 DA - 2022/02 SN - 14 DO - http://doi.org/10.4208/aamm.OA-2020-0400 UR - https://global-sci.org/intro/article_detail/aamm/20277.html KW - Symmetric inertial alternating direction method of multipliers, convergence rate, elliptic equation constraint, finite element method. AB -

A new algorithm, called symmetric inertial alternating direction method of multipliers (SIADMM), is designed for separable convex optimization problems with linear constraints in this paper. The convergence rate of the SIADMM is proved to be $\mathcal{O}(1/ \sqrt{k})$. Two kinds of elliptic equation constrained optimization problems, the unconstrained cases as well as the box-constrained cases of the distributed control and the Robin boundary control, are analyzed theoretically and solved numerically. First, the existence and uniqueness of the solutions to these problems are proved. Second, these continuous optimization problems are transformed into discrete optimization problems by the finite element method, and then the discrete optimization problems are solved by the proposed SIADMM. Numerical experiments with different problems are investigated to demonstrate the efficiency of the SIADMM. And the numerical performance of the SIADMM is better than the performance of the ADMM. Moreover, the numerical results show that the convergence rate of the SIADMM tends to be faster than $\mathcal{O}(1/ \sqrt{k})$ in calculation process.

Mengyue Wu, Wenbao Ai, Jianhua Yuan & Hui Tian. (2022). A Symmetric Inertial Alternating Direction Method of Multipliers for Elliptic Equation Constrained Optimization Problem. Advances in Applied Mathematics and Mechanics. 14 (3). 596-621. doi:10.4208/aamm.OA-2020-0400
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