On the Convergence and O(1/N) Complexity of a Class of Nonlinear Proximal Point Algorithms for Monotonic Variational Inequalities
AbstractThis paper presents a class of proximal point algorithms using a nonlinear proximal term for monotonic variational inequality problems. This work extents proximal point algorithms using Bregman distance for minimization problems, and differs with J. Eckstein's approximate iterations in Bregman-function-based proximal algorithms (1998). We study the convergence of the proposed algorithms and obtain a $O(1/N)$ computing complexity/convergence rate of the algorithms. Further more, connections to some existed popular methods were given, which shows that our algorithm can include these methods within a general form.
A. Auslender, M. Teboulle, Lagrangian duality and related multiplier methods for variational inequality problems, SIAM Journal of Optimization 10(4), 1097-1115(2000)
J. Burke, J. Qian, On the superlinear convergence of the variable metric proximal point algorithm using Broyden and BFGS matrix secant updating, Mathematical Programming. 88(1), 157-181 (2000)
Y. Censor and S. Zenios, Proximal Minimization Algorithm with D-Functions, Journal of Optimization Theory and Applications, 73(3), (1992).
Eckstein, J.: Approximate iterations in Bregman-function-based proximal algorithms. Mathematical Programming, Ser. A 83(1), 113-123(1998)
M. Ferris,J. Pang, Engineering and economic applications of complementarity problems. SIAM Rev. 39(4), 669-713 (1997)
He, B.S.: Inexact implicit methods for monotone general variational inequalities. Mathematical Programming, Ser. A 86(1), 199-217 (1999)
B. He, X. Fu and Z. Jiang, PPA using a linear proximal term, Journal of Optimization Theory and Applications, 141:209-239, 2009.
B. Martinet, Regularisation d’in’equations variationelles par approximations succesives, Rev. Francaise d’Inform. Recherche Operation, 4, 154-159 (1970).
R. Rockafellar, Monotone operators and the PPA, SIAM Journal on Control and Optimization, 14, 877-898 (1976).
M. Teboulle, Convergence of proximal-like algorithms. Journal of Optimization Theory and Applications, 7, 1069-1083(1997)
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).