Improved Strict Contraction Of The Peaceman-Rachford Splitting Algorithm

Convex programming and variational inequality problem play a very important role in many fields such as network economics, traffic equilibrium, statistics applica-tion, data analysis and so on. Consequently, how to design efficient algorithms for solving these problems is a hot research topic.The Peaceman-Rachford splitting method is very efficient for the convex mini-mization problem with linear constraints and a seperable objective function. How-ever, its convergence was not guaranteed without extra requirements. He et al. [18] considered a strictly contractive Peaceman-Rachford splitting method by employing a suitable underdetermined relaxation factor.In this paper, we further extend the so-called strictly contractive Peaceman-Rachford splitting method by using two different relaxation factors. We characterize the relation between these two factors, and show that the second one is allowed to be larger than 1, which can improve the numerical performance of the algorithm. Furthermore, we show that the proposed modified strictly contractive Peaceman-Rachford splitting method is convergent and also prove O(1/t) convergence rate in ergodic and nonergodic sense under suitable conditions, respectively. Further numerical experiments demonstrate the efficiency of the proposed method.