Algorithm Research On Constrained Optimization Problems Governed By Elliptic Equations

We focus on the problem of optimal control.The optimal control problem is such a process that is looking for minmizers or maximizers of a required performance index under some constrains.The optimal control problem is widely used in life,and it belongs to an application of optimization method in mathematics.In this thesis,we investigate three algorithms for solving the state constrained optimization problems governed by elliptic equations,including the alternating direction method of multipliers(ADMM),inertia alternating direction method of multipliers(IADMM)and symmetric alternating direction method of multipliers(SADMM).We study two classes of optimal control models,including unconstrained and box-constrained cases of the diffusion-convection equation control problems.For the two classes of optimal control models,we study the existence and uniqueness with a detailed proof.Then the original optimal control problems are transformed into discrete optimization system by the finite element method,and the convergence error of finite element method is considered.We solve the discrete optimization system by ADMM,IADMM and SADMM,also the process and reason of optimization of IADMM and SADMM are given in detail.We prove the convergence of ADMM,and the convergence rate o(1/k)under the worst case is analyzed.Finally,we show the numerical results of the finite element errors and the convergence rate of the three algorithms through numerical experiments.The numerical results of the finite element errors are basically consistent with the theoretical analysis.For the convergence rate of the three algorithms,SADMM is the best,and ADMM is the last one.The numerical experiments show that the design algorithms are effective and improve the performance of the ADMM,which also illustrate the efficiency of ADMMs' algorithms in solving optimal control problems.