Parametric Predictor-Corrector Splitting Method For Solving Separable Convex Optimization Problems

In recent years,the convex optimization problem has been widely used in mathemat-ical planning,network economy,traffic optimization and image processing.The alternating direction method of multipliers has a good effect on solving a linear convex optimization problem with a separable structure and has become a hot topic in recent years.This method is one of the most effective methods for solving separable convex optimization problems,but for the The separable convex optimization problem for(m ? 3)variables can guarantee the convergence of the algorithm if the objective function is at least m-2 block strong convex function.It is better to keep the advantages of the alternating direction multiplier method.On the basis of this,this paper proposes three new effective parametric prediction-correction splitting algorithms,and can prove the convergence under weaker conditions,and apply it to the practical problems of quadratic programming.For solving the convex optimization problem with three separable variables,Han et al.proposed a partial splitting augmented Lagrangian method.Based on this,this paper propose two new Partial parallel parametric prediction-correction splitting algorithm.In the prediction step of the first new algorithm proposed in this paper,the value of the penalty factor is changed by adding parameters,so that the prediction step is changed.In the correction step,the correction variable is unchanged and only the correction form is changed.The relationship between the parameters and the step size gives the range of values of the parameters.The convergence of the algorithm is proved under certain conditions and within the range of parameter selection,and the partial parallel parameter predictive correction splitting algorithm for the newly proposed correction two variables is a generalization of the partial parallel split augmented Lagrange method of Han et al.The main difference between tlhe second new algorithm proposed and the first new algorithm proposed is that the calibration variables are different,and the rest are the same.And the theory proved that a partial parallel parameter predictive correction splitting algorithm for the newly proposed correction of three variables is a generalization of the partial parallel predictive correction algorithm of Bai and Xu.Finally,the feasibility and effectiveness of the two new algorithms can be derived from numerical tests.Based on the partial parallel splitting and augmenting of the Lagrangian method of Han et al,and based on the ideas of the two previous new algorithms,a new completely parallel inclusion-predictive-correction splitting Algorithm is proposed.In the predic-tion step,the partial parallelism becomes completely parallel,and the prediction step is changed by adding parameters to change the value of the penalty factor.In the correction step,all the variables are corrected and their correction forms are changed at the same time.By considering the relationship between the parameters and the step size,the range of values of the parameters is obtained.The convergence of the algorithm is proved under certain conditions and within the range of parameter selection,and the new proposed Complete parallel inclusion parameter prediction-correction splitting algorithm is a gen-eralization of Xu's complete parallel prediction-correction algorithm.Finally,numerical experiments can be used to derive the feasibility and effectiveness of the new algorithm.