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Abstract
This work is motivated by a recent work on an extended linear proximal point algorithm (PPA) [B.S. He, X.L. Fu, and Z.K. Jiang, Proximal-point algorithm using a linear proximal term, J. Optim. Theory Appl. 141 (2009), pp. 299–319], which aims at relaxing the requirement of the linear proximal term of classical PPA. In this paper, we make further contributions along the line. First, we generalize the linear PPA-based contraction method by using a nonlinear proximal term instead of the linear one. A notable superiority over traditional PPA-like methods is that the nonlinear proximal term of the proposed method may not necessarily be a gradient of any functions. In addition, the nonlinearity of the proximal term makes the new method more flexible. To avoid solving a variational inequality subproblem exactly, we then propose an inexact version of the developed method, which may be more computationally attractive in terms of requiring lower computational cost. Finally, we gainfully employ our new methods to solve linearly constrained convex minimization and variational inequality problems.
Acknowledgements
The authors are grateful to the anonymous referees for their valuable comments on the earlier versions of this paper. The research of H. He is partially supported by the National Natural Science Foundation of China (NSFC) (11301123), the Zhejiang Provincial NSFC (LZ14A010003), and the Research Foundation of Hangzhou Dianzi University (KYS075612037); The research of X. Cai is supported by the NSFC, the Natural Science Foundation of Jiangsu Province (BK20140914); The research of D. Han is supported by the NSFC (11371197); The research of X. Cai and D. Han is also supported by a project funded by PAPD of Jiangsu Higher Education Institutions.