A Symmetric Inertial Alternating Direction Method of Multipliers for Elliptic Equation Constrained Optimization Problem (Advances in Applied Mathematics and Mechanics, 2022)
A new algorithm, called symmetric inertial alternating direction method ofmultipliers (SIADMM), is designed for separable convex optimization problems withlinear 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 andthe 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 optimizationproblems by the finite element method, and then the discrete optimization problemsare solved by the proposed SIADMM. Numerical experiments with different problemsare investigated to demonstrate the efficiency of the SIADMM. And the numerical performance of the SIADMM is better than the performance of the ADMM. Moreover, thenumerical results show that the convergence rate of the SIADMM tends to be fasterthan $\mathcal{O}(1/ \sqrt{k})$ in calculation process.