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 - <p style="text-align: justify;">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.</p>