Predictor-corrector Decomposition Method For Solving Separable Convex Optimization

With the rapid development of science and technology,cloud computing,big data and other areas will also appear.In order to solve the nonlinear programming problem of large-scale,many experts and scholars are devoted to design efficient decomposition algorithms.The decomposition method has the advantages of reducing the dimension,and computing the sub problems in parallel.A lot of decomposition methods for separable convex optimization problem have been developed,such as alternating direction method of multipliers,predictor-corrector proximal multiplier method,parallel splitting augmented lagrangian algorithms,and so on.For separable convex optimization problems with two variables,the alternating di-rection method of multipliers is a relatively mature method for solving such problems.The predictor-corrector proximal multiplier method makes full use of the twice corrections of the multiplier,and possesses good theoretical results.In this paper,the prediction-correction strategy for the Lagrangian multiplier is adopted in the alternating direction method of multipliers to construct a predictor-corrector alternating direction method of multiplier for solving block-separable linearly constrained convex optimization problem.We transform the original convex programming problem into an equivalent variational in-equality problem with separable structure,and under the assumptions that the Lagrangian has a saddle point and the matrices of the linear constraints are full-column-rank,the con-vergence of the algorithm is established.The second algorithm is proposed,which is the parallel predictor-corrector alternating direction method of multiplier.This algorithm uses the mind of the parallel splitting augmented lagrangian algorithms.In the iteration of the algorithm,the variables are calculated in parallel.For convex optimization problems with m(m?3)variables,the convergence of the directly extended ADMM remains open.The latest literature shows that under the as-sumption of at least m-2 strongly convex functions,the convergence can be established.In this thesis,we propose the extensions of the predictor-corrector alternating direction methods of multiplier which incorporate the non-parallel and the parallel structures re-spectively.The convergence is established without the assumption of strongly convexity.The effectiveness of the proposed algorithms is illustrated by some numerical examples.