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Convergence Analysis of Gauss-Type Proximal Point Method for Variational Inequalities
Current Issue
Volume 2, 2014
Issue 1 (February)
Pages: 5-14   |   Vol. 2, No. 1, February 2014   |   Follow on         
Paper in PDF Downloads: 22   Since Aug. 28, 2015 Views: 1649   Since Aug. 28, 2015
Authors
[1]
Mohammed Harunor Rashid , Department of Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh.
Abstract
In the present paper, we introduce a Gauss-type proximal point algorithm for solving the variational inequality problem, that is, a problem involving functional acting between Banach space and its dual space, which has to be solved for all possible values of a given variable belonging usually to a convex set. We establish the convergence criteria of the Gauss-type proximal point algorithm, which guarantees the existence and the convergence of any sequence generated by this algorithm under mild conditions. More precisely, semilocal and local convergences of the Gauss-type proximal point algorithm are analyzed.
Keywords
Variational inequality, Metrically regular mappings, Proximal point algorithm, Semi-local convergence
Reference
[1]
B. Martinet, Regularisation d'inequations variationnelles par approximations successives, Rev. Fr. Inform Rech. Oper. 3(1970) 154--158.
[2]
R.T. Rockafellar, R. J-B. Wets, Variational Analysis, A Series of Comprehensive Studies in Mathematics, Vol. 317, Springer, 1998.
[3]
R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14(1976) 877--898.
[4]
R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res. 1(1976) 97--116.
[5]
J. E. Spingarn, Submonotone mappings and the proximal point algorithm, Numer. Funct. Anal. Optim. 4(1982) 123--150.
[6]
M. V. Solodov, B. F. Svaiter, A hybrid projection-proximal point algorithm, J. Convex Anal. 6(1)(1999) 59--70.
[7]
N. Xiu, J. Zhang, On finite convergence of proximal point algorithms for variational inequalities, J. Math. Annl. Appl. 312(2005) 148--158.
[8]
A. Auslender, M. Teboulle, Lagrangian duality and related multiplier methods for variational inequality problems, SIAM J. Optim. 10(2000) 1097--1115.
[9]
A. Kaplan, R. Tichatschke, Proximal-based regularization methods and succesive approximation of variational inequalities in Hilbert spaces, Control and Cybernetics 31(3)(2002) 521--544.
[10]
Corina L Chiriac, Proximal point methods for variational inequalities involving regular mappings, Romai J. 6(1) (2010) 41--45.
[11]
A. L. Dontchev, R. T. Rockafellar, Regularity and conditioning of solution mappings in variational analysis, Set-valued Anal. 12(2004) 79--109.
[12]
A. L. Dontchev, R. T. Rockafellar, Implicit functinos and solution mappings: A view from variational analysis, Dordrecht, Heidelberg, London, LLC, New York, Springer, 2009.
[13]
M. H. Rashid, J. H. Jinhua, C. Li, Convergence Analysis of Gauss-type proximal point method for metric regular mappings, J. Nonlinear conv. Anal. 14(3) (2013) 627--635.
[14]
B. S. Mordukhovich, Sensitivity analysis in nonsmooth optimization: Theoretical Aspects of Industrial Design (D. A. Field and V. Komkov, eds.), SIAM Proc. Appl. Math. 58(1992) 32--46.
[15]
J. P. Penot, Metric regularity, openness and Lipschitzian behavior of multifunctions, Nonlinear Anal. 13(1989) 629--643.
[16]
J. P. Aubin, Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res. 9(1984) 87--111.
[17]
J. P. Aubin, H. Frankowska, Set-valued analysis, Birkhauser, Boston, 1990.
[18]
B. S. Mordukhovich, Variational Analysis and generalized differentiation I: Basic theory, Grundlehren Math. Wiss. Vol. 330, Springer-Verlag, Berlin, 2006.
[19]
A. L. Dontchev, A. S. Lewis, R. T. Rockafellar, The radius of metric regularity, Trans AMS 355(2002) 493--517.
[20]
A. L. Dontchev, W. W. Hager, An inverse mapping theorem for set-valued maps, Proc. Amer. Math. Soc. 121(1994) 481--498.
[21]
A. D. Ioffe, V. M.Tikhomirov, Theory of extremal problems, Studies in Mathematics and its Applications, Amsterdam, New York, North-Holland, 1979.
[22]
F. J. Aragon Artacho, M. H. Geoffroy, Uniformity and inexact version of a proximal point method for metrically regular mappings, J. Math. Anal. Appl. 335(2007) 168--183.
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