A New Approximation Method for Finding Zeros of Maximal Monotone Operators
Main Article Content
Abstract
One of the major problems in the theory of maximal monotone operators is to find a point in the solution set Zer( ), set of zeros of maximal monotone mapping . The problem of finding a zero of a maximal monotone in real Hilbert space has been investigated by many researchers. Rockafellar considered the proximal point algorithm and proved the weak convergence of this algorithm with the maximal monotone operator. Güler gave an example showing that Rockafellar’s proximal point algorithm does not converge strongly in an infinite-dimensional Hilbert space. In this paper, we consider an explicit method that is strong convergence in an infinite-dimensional Hilbert space and a simple variant of the hybrid steepest-descent method, introduced by Yamada. The strong convergence of this method is proved under some mild conditions. Finally, we give an application for the optimization problem and present some numerical experiments to illustrate the effectiveness of the proposed algorithm.
Keywords
Zero points, variational inequalities, maximal monotone operators, strongly monotone operators, lipschitz continuous operators
Article Details
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References
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vol 54, no 4, pp 1347-1359, 2017.
http://www.koreascience.or.kr/article/JAKO20172383
9750277.page
proximal point algorithm, SIAM J. Control Optim., vol.
14, no. 5, pp. 877-898, 1976.
https://doi.org/10.1137/0314056
[2]. Takahashi W., Convex Analysis and Approximation of
Fixed Points, Yokohama Publishers, Yokohama, 2000
(4-946552-04-9: Japanese).
[3]. Yamada Y., The hybrid steepest-descent method for
variational inequalities problems over the intersection
of the fixed point sets of nonexpansive mappings,
Edited by Butnariu D., Censor Y., Reich S., Inhently
Parallel Algorithms in Feasibility and Optimization and
Their Applications, North-Holland, Amsterdam,
Holland, pp. 473-504, 2001.
https://doi.org/10.1016/S1570-579X(01)80028-8
[4]. Ceng L.C., Ansari Q.H., Yao J.C., Mann-type steepset
descent and modified hybrid steepset-descent methods
for variational inequality in Banach spaces, Numer.
Funct. Anal. Optim., vol. 29, pp. 987-1033, 2008.
https://doi.org/10.1080/01630560802418391
[5]. Opial Z., Weak convergence of the sequence of
successive approximations for nonexpansive mappings,
Bull. Am. Math. Soc., vol. 73, no. 4, pp. 591-597, 1967.
https://doi.org/10.1090/S0002-9904-1967-11761-0
[6]. Xu H.K., Inequalities in Banach spaces with
applications, Nonlinear Anal., vol. 16, no. 12, pp. 1127
1138, 1991.
https://doi.org/10.1016/0362-546X(91)90200-K
[7]. Reich S., Extension problems for accretive sets in
Banach spaces, J. Funct. Anal., vol. 26, pp. 378-395,
1977.
https://doi.org/10.1016/0022-1236(77)90022-2
[8]. Tuyen T.M., Thuy Ng.T.T., Trang Ng.M., A strong
convergence theorem for a parallel iterative method for
solving the split common null point problem in Hilbert
spaces, J. Optim. Theory Appl., vol. 183, no. 2, pp. 271
291, 2019.
https://doi.org/10.1007/s10957-019-01523-w
[9]. Maingé P.E., Strong convergence of projected
subgradient methods for nonsmooth and nonstrictly
convex minimization, Set-Valued Anal., vol. 16, no. 7-
8, pp. 899-912, 2008.
https://doi.org/10.1007/s11228-008-0102-z
[10]. Xu H.K., Strong convergence of an iterative method for
nonexpansive and accretive operators, J. Math. Anal.
Appl., vol. 314, pp. 631- 643, 2006.
https://doi.org/10.1016/j.jmaa.2005.04.082
[11]. Kinderlerhrer D., Stampacchia G., An Introduction to
Variational Inequalities and Their Applications,
Academic Press, New York, 1980, pp. 22-104.
https://epubs.siam.org/doi/book/10.1137/1.978089871
9451?mobileUi=0
[12]. Kim J.K., Tuyen T.M., New Iterative methods for
finding a common zero of a finite family of monotone
operators in Hilbert spaces, Bull. Korean Math. Soc.,
vol 54, no 4, pp 1347-1359, 2017.
http://www.koreascience.or.kr/article/JAKO20172383
9750277.page