NON-STRICTLY CONVEX MINIMIZATION OVER THE FIXED POINT SET OF AN ASYMPTOTICALLY SHRINKING NONEXPANSIVE MAPPING
Suppose that T is a nonexpansive mapping on a real Hilbert space satisfying for some R > 0. Suppose also that a mapping is κ-Lipschitzian over and paramonotone over . Then it is shown that a variation of the hybrid steepest descent method (Yamada, Ogura, Yamashita and Sakaniwa (1998), Deutsch and...
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| Published in | Numerical functional analysis and optimization Vol. 23; no. 1-2; pp. 113 - 137 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Taylor & Francis Group
05.01.2002
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0163-0563 1532-2467 |
| DOI | 10.1081/NFA-120003674 |
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| Abstract | Suppose that T is a nonexpansive mapping on a real Hilbert space
satisfying
for some R > 0. Suppose also that a mapping
is κ-Lipschitzian over
and paramonotone over
. Then it is shown that a variation of the hybrid steepest descent method (Yamada, Ogura, Yamashita and Sakaniwa (1998), Deutsch and Yamada (1998) and Yamada (1999-2001)):
generates a sequence (u
n
) satisfying
, when
is finite dimensional, where
for all
is the solution set of the variational inequality problem
. This result relaxes the condition on
and (λ
n
) of the hybrid steepest descent method (Yamada (2001)), and makes the method applicable to the significantly wider class of convexly constrained inverse problems as well as the non-strictly convex minimization over the fixed point set of asymptotically shrinking nonexpansive mapping. |
|---|---|
| AbstractList | Suppose that T is a nonexpansive mapping on a real Hilbert space
satisfying
for some R > 0. Suppose also that a mapping
is κ-Lipschitzian over
and paramonotone over
. Then it is shown that a variation of the hybrid steepest descent method (Yamada, Ogura, Yamashita and Sakaniwa (1998), Deutsch and Yamada (1998) and Yamada (1999-2001)):
generates a sequence (u
n
) satisfying
, when
is finite dimensional, where
for all
is the solution set of the variational inequality problem
. This result relaxes the condition on
and (λ
n
) of the hybrid steepest descent method (Yamada (2001)), and makes the method applicable to the significantly wider class of convexly constrained inverse problems as well as the non-strictly convex minimization over the fixed point set of asymptotically shrinking nonexpansive mapping. |
| Author | Ogura, Nobuhiko Yamada, Isao |
| Author_xml | – sequence: 1 givenname: Nobuhiko surname: Ogura fullname: Ogura, Nobuhiko organization: Precision and Intelligence Laboratory , Tokyo Institute of Technology – sequence: 2 givenname: Isao surname: Yamada fullname: Yamada, Isao organization: Department of Communications and Integrated Systems , Tokyo Institute of Technology |
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| Cites_doi | 10.1109/36.298007 10.1007/978-94-011-4066-9 10.1090/S0002-9904-1967-11761-0 10.1080/01630569808816822 10.1109/5.214546 10.1073/pnas.54.4.1041 10.1007/BF01891408 10.1109/78.709518 10.1137/0909048 10.1090/conm/204/02620 10.1080/01630569808816813 10.1007/978-1-4612-4838-5 10.1016/0021-9045(92)90117-7 10.1080/01630569208816489 10.1007/BF01890024 10.1137/0802021 10.1007/978-1-4612-5020-3 10.1109/TMI.1982.4307555 10.1109/29.32281 10.1109/78.782189 10.1137/S0036144593251710 10.1109/TASSP.1983.1164011 |
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| References | Yamada I. (CIT3674-26) 2001 CIT3674-25 CIT3674-29 Barbu V. (CIT3674-1) 1986 CIT3674-4 Takahashi W. (CIT3674-24) 2000 Censor Y. (CIT3674-6) 1998; 81 CIT3674-2 Stark H. (CIT3674-23) 1998 Ekeland I. (CIT3674-14) 1976 CIT3674-8 Yamada I. (CIT3674-27) 2000; 83 CIT3674-30 CIT3674-9 CIT3674-11 CIT3674-10 CIT3674-13 Zeidler E. (CIT3674-33) 1985 CIT3674-12 Zeidler E. (CIT3674-32) 1990 CIT3674-17 CIT3674-16 Bauschke H.H. (CIT3674-3) 1997; 204 Censor Y. (CIT3674-7) 1997 CIT3674-19 CIT3674-18 Yamada I. (CIT3674-28) 1999; 5 Zeidler E. (CIT3674-31) 1986 Goebel K. (CIT3674-15) 1984 CIT3674-20 Butnariu D. (CIT3674-5) 2000 CIT3674-22 CIT3674-21 |
| References_xml | – ident: CIT3674-16 doi: 10.1109/36.298007 – volume: 5 start-page: 37 year: 1999 ident: CIT3674-28 publication-title: Proc. of 1999 IEEE International Symposium on Circuits and Systems – volume-title: Totally Convex Functions for Fixed Point Computation and Infinite Dimensional Optimization year: 2000 ident: CIT3674-5 doi: 10.1007/978-94-011-4066-9 – ident: CIT3674-20 doi: 10.1090/S0002-9904-1967-11761-0 – ident: CIT3674-29 doi: 10.1080/01630569808816822 – ident: CIT3674-10 doi: 10.1109/5.214546 – volume-title: Nonlinear Functional Analysis–-Fixed Point Theory and Its Applications year: 2000 ident: CIT3674-24 – volume: 81 start-page: 373 year: 1998 ident: CIT3674-6 publication-title: Math. Programming – ident: CIT3674-4 doi: 10.1073/pnas.54.4.1041 – ident: CIT3674-8 doi: 10.1007/BF01891408 – ident: CIT3674-22 doi: 10.1109/78.709518 – ident: CIT3674-19 doi: 10.1137/0909048 – volume: 204 start-page: 1 year: 1997 ident: CIT3674-3 publication-title: Contemp. Math. doi: 10.1090/conm/204/02620 – ident: CIT3674-12 doi: 10.1080/01630569808816813 – volume-title: Nonlinear Functional Analysis and Its Applications, I–-Fixed Point Theorems year: 1986 ident: CIT3674-31 doi: 10.1007/978-1-4612-4838-5 – volume-title: Inherently Parallel Algorithm for Feasibility and Optimization year: 2001 ident: CIT3674-26 – volume-title: Convexity and Optimization in Banach Spaces, year: 1986 ident: CIT3674-1 – ident: CIT3674-9 doi: 10.1016/0021-9045(92)90117-7 – volume-title: Vector Space Projection–-A Numerical Approach to Signal and Image Processing. Neural Nets and Optics year: 1998 ident: CIT3674-23 – ident: CIT3674-13 doi: 10.1080/01630569208816489 – ident: CIT3674-18 doi: 10.1007/BF01890024 – volume-title: Nonlinear Functional Analysis and Its Applications, II/B–-Nonlinear Monotone Operator year: 1990 ident: CIT3674-32 – ident: CIT3674-25 doi: 10.1137/0802021 – volume-title: Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings year: 1984 ident: CIT3674-15 – volume-title: Nonlinear Functional Analysis and Its Applications, III–-Variational Methods and Optimization year: 1985 ident: CIT3674-33 doi: 10.1007/978-1-4612-5020-3 – ident: CIT3674-30 doi: 10.1109/TMI.1982.4307555 – ident: CIT3674-21 doi: 10.1109/29.32281 – ident: CIT3674-11 doi: 10.1109/78.782189 – ident: CIT3674-2 doi: 10.1137/S0036144593251710 – volume-title: Parallel Optimization: Theory, Algorithm and Optimization year: 1997 ident: CIT3674-7 – volume-title: Convex Analysis and Variational Problems year: 1976 ident: CIT3674-14 – volume: 83 start-page: 616 year: 2000 ident: CIT3674-27 publication-title: The Journal of the IEICE (in Japanese) – ident: CIT3674-17 doi: 10.1109/TASSP.1983.1164011 |
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| Snippet | Suppose that T is a nonexpansive mapping on a real Hilbert space
satisfying
for some R > 0. Suppose also that a mapping
is κ-Lipschitzian over
and paramonotone... |
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| StartPage | 113 |
| SubjectTerms | 2000 Mathematics Subject Classification Convex optimization Convex projection Fixed point theorem Generalized convex feasible set Inverse problem Monotone operator Nonexpansive mapping Steepest descent method Variational inequality problem |
| Title | NON-STRICTLY CONVEX MINIMIZATION OVER THE FIXED POINT SET OF AN ASYMPTOTICALLY SHRINKING NONEXPANSIVE MAPPING |
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