Research On Iterative Methods For Solving Equality Constrained Convex Optimization Problems

With the rapid development of science and technology,there are many optimization problems in scientific computing and engineering applications.The equality-constrained convex optimization problem for separable variables is a very important optimization problem,which has important theoretical significance and applicable value.The alternating direction method of multipliers（ADMM）is a very effective method to solve this problem.However,its convergence rate is relatively slow,and the stopping criterion is difficult to grasp,and it often takes a long time to solve this optimization problem.Based on this facts,the different modified ADMM methods are proposed by some scholars for different forms of convex optimization problems.Based on the existing modified ADMM methods,this paper proposes two modified ADMM methods for two different forms of equality-constrained convex optimization problems,respectively.The main research work is as follows:Firstly,considering a class of relatively simple equality-constrained convex optimization problem for separable variables—the equality-constrained quadratic programming problem for separable variables,this paper constructs a preconditioned and proximal alternating direction method of multipliers（PPADMM）utilizing matrix preconditioning strategy and parameter accelerating technique to modify the ADMM method,which can effectively improve the convergence rate of the ADMM method.Based on strictly matrix analysis,this paper proves that this method is asymptotically convergent,and gives its convergence factor.Then,aiming at the general equality-constrained convex optimization problem for separable variables,a generalized preconditioned and proximal alternating direction method of multipliers（PPADMM）is proposed.It is proved theoretically that the method is convergent,and the worst-case convergence rate is O（1/n）,and the convergence conditions of the GPPADMM method are discussed in the case of W^-1=Q.Last but not least,the numerical examples show the PPADMM and GPPADMM methods are efficient,stable and flexible for solving the equality-constrained quadratic programming problem for separable variables.However,the GPPADMM method has a faster convergence rate than the PPADMM method,and the GPPADMM method is so applicable that it can be used to solve the optimization problem where the objective function is the general convex function.Nevertheless,the PPADMM method is only suitable for solving the quadratic programming problem.