| Block separable convex optimization problems with linear constraints often appear in many fields such as multitask learning,image processing,engineering management,compressed sensing,signal denoising and so on.How to solve such problems have aroused a wide range of attention of scholars.The objective function of block separable convex optimization problem with linear constraints is the sum of multiple convex functions.There is no cross term for variables involved in each convex function.A kind of effective method to deal with this problem is the Alternating Direction Method of Multiplier-s(ADMM method).The method is derived from the Augmented Lagrange Multiplier method(ALM)and the Proximity Point Algorithm(PPA).Although the Alternating Di-rection Method of Multipliers for the block separable convex optimization problem with two block variables,it has convergence.However,for the block separable convex opti-mization problem with three or more variables,the convergence of the method can not be guaranteed.Only if the objective function is m-2 strong convex functions,the algorithm can guarantee convergence.This m represents the number of variables.On the basis of the alternating direction method of multipliers,many scholars have taken the point obtained by this method as a prediction point.Then the new iteration point is corrected.Finally,it can be concluded that this type of algorithm has convergence.This algorithms is the prediction-correction decomposition algorithm.In this paper,we consider the solution of convex optimization problem with linear constraint and objective function of three blocks.To solve this problem in the framework of prediction-correction decomposition algorithm,three new algorithms this paper proposed,namely prediction-correction decomposition algorithm based on ADMM(NEW1),prediction-correction decomposition algorithm for local parallel(NEW2)and prediction-correction decomposition algorithm for fully par-allel(NEW3).And under appropriate assumptions,it is proved that these algorithms have convergence.This paper gives numerical examples to show the effectiveness of these algorithms.The main contents of this paper are as follows:In the first chapter,the research background of separable convex optimization prob-lem with linear constraints and two algorithms to describe the problem are introduced.The main work of this paper is also briefly described.The second chapter is a brief introduction,including some common symbols,lemmas,definition and so on.The third chapter proposes a prediction-correction decomposition algorithm based on ADMM(NEW1)under the framework of the prediction-correction decomposition al-gorithm.At the same time,the proof of convergence of the algorithm is given.Finally,numerical experiments are carried out,and the effectiveness of the algorithm is illustrated on the basis of the numerical results of an example.The fourth chapter proposes a prediction-correction decomposition algorithm for lo-cal parallel(NEW2)under the framework of the prediction-correction decomposition al-gorithm.At the same time,the proof of convergence of the algorithm is given.Finally,numerical experiments are carried out,and the effectiveness of the algorithm is illustrated on the basis of the numerical results of an example.The fifth chapter proposes a prediction-correction decomposition algorithm for fully parallel(NEW3).At the same time,the proof of convergence of the algorithm is given.Finally,numerical experiments are carried out,and the effectiveness of the algorithm is illustrated on the basis of the numerical results of an example.The sixth chapter is a summary of the research work of this paper.At the same time,the future research work is prospected. |
