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Extension of the hybrid steepest descent method to a class of variational inequalities and fixed point problems with nonself-mappings. (English) Zbl 1159.65005
Consider \(H\) a Hilbert space, \(D\subset H\) a closed convex set, \(F,T:D\rightarrow H\) with \(\text{Fix}(T):=\{x\in D\mid Tx=x\}\neq\emptyset.\) The aim of the paper is to establish an algorithm to solve the problem VIP\(F,\text{Fix}(T)\): find \(x_{\ast}\in Fix(T)\) such that \(\left\langle F(x_{\ast}),v-x_{\ast}\right\rangle \geq0\) for all \(v\in \text{Fix}(T).\) For this one considers the iteration \(x_{n+1}:=P_{D}T_{w_{n}} P_{D}x_{n}-\alpha_{n}F(P_{D}T_{w_{n}}P_{D}x_{n}),\) where \((\alpha_{n} )\subset[0,1],\) \((w_{n})\subset(0,\infty),\) \(T_{w_{n}}:=(1-w_{n} )I+w_{n}T,\) \(I\) being the identity mapping on \(D\) and \(P_{D}\) being the metric projection from \(H\) to \(D.\) Under some additional assumptions on \(F,T,\) \((\alpha_{n})\) and \((w_{n})\) it is shown that the sequence \((x_{n})\) converges strongly to the (unique) solution VIP\(F,\text{Fix}(T)\).

MSC:
65C20 Probabilistic models, generic numerical methods in probability and statistics
90C29 Multi-objective and goal programming
90C26 Nonconvex programming, global optimization
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[1] DOI: 10.1016/0022-247X(67)90085-6 · Zbl 0153.45701 · doi:10.1016/0022-247X(67)90085-6
[2] DOI: 10.1287/moor.26.2.248.10558 · Zbl 1082.65058 · doi:10.1287/moor.26.2.248.10558
[3] DOI: 10.1109/83.563316 · doi:10.1109/83.563316
[4] DOI: 10.1109/TIP.2004.832922 · Zbl 05452844 · doi:10.1109/TIP.2004.832922
[5] DOI: 10.1080/01630569808816813 · Zbl 0913.47048 · doi:10.1080/01630569808816813
[6] Ekeland I., Convex Analysis and Variational Problems 28 (1999) · Zbl 0939.49002
[7] DOI: 10.1017/CBO9780511526152 · doi:10.1017/CBO9780511526152
[8] DOI: 10.1016/0022-247X(77)90076-2 · Zbl 0361.65057 · doi:10.1016/0022-247X(77)90076-2
[9] Iiduka H., J. Convex Analysis 11 pp 69– (2004)
[10] Itoh S., Pacific J. Math. 79 pp 493– (1978)
[11] DOI: 10.1007/s11117-007-2066-x · Zbl 1145.90052 · doi:10.1007/s11117-007-2066-x
[12] Maingé P.E., Pacific J. Optimization 3 pp 529– (2007)
[13] DOI: 10.1016/j.jmaa.2005.05.028 · Zbl 1095.47038 · doi:10.1016/j.jmaa.2005.05.028
[14] DOI: 10.1016/j.jmaa.2006.06.055 · Zbl 1116.47053 · doi:10.1016/j.jmaa.2006.06.055
[15] Maruster S., Proc. Amer. Math. Soc. 63 pp 69– (1997)
[16] C. Moore ( 1998 ). Iterative aproximation of fixed points of demicontractive maps . The Abdus Salam. Intern. Centre for Theoretical Physics, Trieste, Italy, Scientific Report, IC/98/214, November .
[17] DOI: 10.1081/NFA-120020250 · Zbl 1034.47043 · doi:10.1081/NFA-120020250
[18] Takahashi W., Nonlinear Functional Analysis (2000) · Zbl 0997.47002
[19] Yamada I., Inherently Parallel Algorithm for Feasibility and Optimization and Their Applications pp 473– (2001) · doi:10.1016/S1570-579X(01)80028-8
[20] DOI: 10.1081/NFA-200045815 · Zbl 1095.47049 · doi:10.1081/NFA-200045815
[21] Yamada I., Inverse Problems, Image Analysis and Medical Imaging 313 pp 269– (2002) · doi:10.1090/conm/313/05379
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