A Relaxed Inertial Proximal Peaceman-Rachford Splitting Method For Separable Convex Programming

Convex optimization problems and variational inequality problems have been widely applied in mathematical programming,data analysis,traf-fic optimization and image processing.As a class of convex optimization problems with special structures,the applications of separable convex opti-mization problem in compressed sensing,matrix analysis,image processing,machine learning and many other practical problems are very extensive.Solving these problems effectively is a key step in these practical applica-tions.Therefore,research on how to design an efficient algorithm to solve this kind of problem is of great practical significance.The strictly contractive Peaceman-Rachford(PR)splitting method is an effective method for solving separable convex optimization problem,and the inertial proximal PR splitting method is one of its important variants.Previous studies show that the convergence of the inertial proximal PR s-plitting method can be sufficiently ensured if the relaxation factors in the Lagrangian multiplier updating are underdetermined.Although the un-derdetermined relaxation factors play an important role in ensuring the convergence of the inertial proximal PR splitting method,the introduction of the underdetermined relaxation factors also means that the step-sizes for the Lagrangian multiplier updates are shrunk conservatively.However,small steps decelerate the convergence numerically.Hence,it is of signifi-cance to select large steps as far as possible while the convergence of the algorithm can be still ensuredIn this paper,we propose a relaxed inertial proximal PR splitting method based on the inertial proximal PR splitting method.The new method has a larger feasible set of the relaxation factors,which reduces the number of iterative steps and improves the numerical performance of the algorithm Under the same conditions as the existing inertial proximal PR splitting method,we firstly prove the convergence of the sequence of the function values of the new algorithm and the asymptotic feasibility of the iterative sequences,and then establish the global convergence of the iterative seque-nces generated by the new algorithm.In addition,the asymptotic conver-gence rate of the new algorithm is also established.Numerical experimental results on a sparse signal recovery problem in compressive sensing and a total variation based image denoising problem demonstrate that the new method has some advantages both in iteration steps and iteration time when choosing lager step sizes.