Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Best Proximity Points Kapitel lesen Erstes Kapitel lesen

2014 | OriginalPaper | Buchkapitel

Triple Hierarchical Variational Inequalities

verfasst von : Qamrul Hasan Ansari, Lu-Chuan Ceng, Himanshu Gupta

Erschienen in: Nonlinear Analysis

Verlag: Springer India

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

In this chapter, we give a survey on hierarchical variational inequality problems and triple hierarchical variational inequality problems. By combining hybrid steepest descent method, Mann’s iteration method, and projection method, we present a hybrid iterative algorithm for computing a fixed point of a pseudo-contractive mapping and for finding a solution of a triple hierarchical variational inequality in the setting of real Hilbert space. We prove that the sequence generated by the proposed algorithm converges strongly to a fixed point which is also a solution of this triple hierarchical variational inequality problem. On the other hand, we also propose another hybrid iterative algorithm for solving a class of triple hierarchical variational inequality problems concerning a finite family of pseudo-contractive mappings in the setting of real Hilbert spaces. Under very appropriate conditions, we derive the strong convergence of the proposed algorithm to the unique solution of this class of problems.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Vorheriges Kapitel Hierarchical Minimization Problems and Applications
Nächstes Kapitel Split Feasibility and Fixed Point Problems
Literatur
1.
Ansari, Q.H., Lalitha, C.S., Mehta, M.: Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization. CRC Press, Taylor & Francis Group, Boca Raton (2014)MATH
2.
Auslender, A., Teboulle, M.: Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, New York (2003)MATH
3.
Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities: Applications to Free Boundary Problems. Wiley, Chichester (1984)MATH
4.
Bauschke, H.: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 202, 150–159 (1996)CrossRefMATHMathSciNet
5.
Browder, F.E.: Existence and approximation of solutions of nonlinear variational inequalities. Proc. Natl. Acad. Sci. U.S.A. 56, 1080–1086 (1966)CrossRefMATHMathSciNet
6.
Browder, F.E.: Convergence of approximates to fixed points of nonexpansive nonlinear mappings in Banach spaces. Arch. Ration. Mech. Anal. 24, 82–90 (1967)CrossRefMATHMathSciNet
7.
Browder, F.E., Petryshyn, W.V.: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 20, 197–228 (1967)CrossRefMATHMathSciNet
8.
Cabot, A.: Proximal point algorithm controlled by a slowly vanishing term: applications to hierarchical minimization. SIAM J. Optim. 15, 555–572 (2005)CrossRefMATHMathSciNet
9.
Ceng, L.C., Ansari, Q.H., Wen, C.-F.: Hybrid steepest descent viscosity method for triple hierarchical variational inequalities. Abstr. Appl. Anal. 2012, Article ID 907105 (2012)
10.
Ceng, L.C., Ansari, Q.H., Wong, N.-C., Yao, Y.C.: Implicit iterative method for hierarchical variational inequalities. J. Appl. Math. 2012, Article ID 472935 (2012)
11.
Ceng, L.C., Ansari, Q.H., Yao, Y.C.: Iterative method for triple hierarchical variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 151, 482–512 (2011)CrossRefMathSciNet
12.
Ceng, L.C., Ansari, Q.H., Yao, J.C.: Relaxed hybrid steepest descent method with variable parameters for triple hierarchical variational inequality. Appl. Anal. 91, 1793–1810 (2012)CrossRefMATHMathSciNet
13.
Ceng, L.C., Wen, C.F.: Hybrid steepest-descent methods for triple hierarchical variational inequalities. Taiwan. J. Math. 17, 1441–1472 (2013)CrossRefMATHMathSciNet
14.
Ceng, L.C., Xu, H.K., Yao, J.C.: A hybrid steepest-descent method for variational inequalities in Hilbert spaces. Appl. Anal. 87, 575–589 (2008)CrossRefMATHMathSciNet
15.
Chang, S.S.: Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 323, 1402–1416 (2006)CrossRefMATHMathSciNet
16.
Chidume, C.E., Zegeye, H.: Approximate fixed point sequences and convergence theorems for Lipschitz pseudo-contractive maps. Proc. Am. Math. Soc. 132, 831–840 (2003)CrossRefMathSciNet
17.
Cianciaruso, F., Colao, V., Muglia, L., Xu, H.K.: On an implicit hierarchical fixed point approach to variational inequalities. Bull. Austr. Math. Soc. 80, 117–124 (2009)CrossRefMATHMathSciNet
18.
Combettes, P.L.: A block-iterative surrogate constraint splitting method for quadratic signal recovery. IEEE Trans. Signal Process 51, 1771–1782 (2003)CrossRefMathSciNet
19.
Combettes, P.L., Hirstoaga, S.A.: Approximating curves for nonexpansive and monotone operators. J. Convex Anal. 13, 633–646 (2006)
20.
21.
Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I, II. Springer, New York (2003)
22.
Fichera, G.: Problemi elettrostatici con vincoli unilaterali; il problema di Signorini con ambigue condizioni al contorno. Atti Acad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I 7, 91–140 (1964)
23.
Geobel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics, vol. 28. Cambridge University Press, Cambridge (1990)
24.
Glowinski, R., Lions, J.L., Trémolières, R.: Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam (1981)MATH
25.
Goh, C.J., Yang, X.Q.: Duality in Optimization and Variational Inequalities. Taylor & Francis, London (2002)CrossRefMATH
26.
Hartmann, P., Stampacchia, G.: On some nonlinear elliptic differential functional equations. Acta Math. 115, 271–310 (1966)CrossRefMathSciNet
27.
Iiduka, H.: Strong convergence for an iterative method for the triple-hierarchical constrained optimization problem. Nonlinear Anal. 71, e1292–e1297 (2009)CrossRefMATHMathSciNet
28.
Iiduka, H.: A new iterative algorithm for the variational inequality problem over the fixed point set of a firmly nonexpansive mapping. Optimization 59, 873–885 (2010)CrossRefMATHMathSciNet
29.
Iiduka, H.: Iterative algorithm for solving triple-hierarchical constrained optimization problem. J. Optim. Theory Appl. 148, 580–592 (2011)CrossRefMATHMathSciNet
30.
31.
Iiduka, H.: Iterative algorithm for triple-hierarchical constrained nonconvex optimization problem and its application to network bandwidth allocation. SIAM J. Optim. 22, 862–878 (2012)CrossRefMATHMathSciNet
32.
Iiduka, H., Takahashi, W., Toyoda, M.: Approximation of solutions of variational inequalities for monotone mappings. Panam. Math. J. 14, 49–61 (2004)MATHMathSciNet
33.
Iiduka, H., Uchida, M.: Fixed point optimization algorithms for network bandwidth allocation problems with compoundable constraints. IEEE Commun. Lett. 15, 596–598 (2011)
34.
Iiduka, H., Yamada, I.: A use of conjugate gradient direction for the convex optimization problem over the fixed point set of a nonexpansive mapping. SIAM J. Optim. 19, 1881–1893 (2009)CrossRefMATHMathSciNet
35.
Isac, G.: Topological Methods in Complementarity Theory. Kluwer Academic Publishers, Dordrecht (2000)CrossRefMATH
36.
Jitpeera, T., Kumam, P.: A new explicit triple hierarchical problem over the set of fixed points and generalized mixed equilibrium problems. J. Inequal. Appl. 2012, Article ID 82 (2012)
37.
Jitpeera, T., Kumam, P.: Algorithms for solving the variational inequality problem over the triple hierarchical problem. Abstr. Appl. Anal. 2013, Article ID 827156 (2013)
38.
Jung, J.S.: Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 302, 509–520 (2005)CrossRefMATHMathSciNet
39.
40.
Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and their Applications. Academic Press, New York (1980)MATH
41.
Kong, Z.R., Ceng, L.C., Ansari, Q.H., Pang, C.T.: Multistep hybrid extragradient method for triple hierarchical variational inequalities. Abstr. Appl. Anal. 2013, Article ID 718624 (2013)
42.
Konnov, I.V.: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2001)CrossRefMATH
43.
Konnov, I.V.: Equilibrium Models and Variational Inequalities. Elsevier, Amsterdam (2007)MATH
44.
Lions, J.-L., Stampacchia, G.: Inéquations variationnelles non coercives. C.R. Math. Acad. Sci. Paris 261, 25–27 (1965)
45.
46.
Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, New York (1996)CrossRef
47.
Lu, X., Xu, H.-K., Yin, X.: Hybrid methods for a class of monotone variational inequalities. Nonlinear Anal. 71, 1032–1041 (2009)CrossRefMATHMathSciNet
48.
Mainge, P.E., Moudafi, A.: Strong convergence of an iterative method for hierarchical fixed-point problems. Pac. J. Optim. 3, 529–538 (2007)MATHMathSciNet
49.
Mancio, O., Stampacchia, G.: Convex programming and variational inequalities. J. Optim. Theory Appl. 9, 2–23 (1972)
50.
Marino, G., Xu, H.K.: A general iterative methods for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 318, 43–52 (2006)CrossRefMATHMathSciNet
51.
Marino, G., Xu, H.K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 329, 336–349 (2007)CrossRefMATHMathSciNet
52.
Marino, G., Xu, H.K.: Explicit hierarchical fixed point approach to variational inequalities. J. Optim. Theory Appl. 149, 61–78 (2011)CrossRefMATHMathSciNet
53.
Matinez-Yanes, C., Xu, H.K.: Strong convergence of the CQ method for fixed point processes. Nonlinear Anal. 64, 2400–2411 (2006)CrossRefMathSciNet
54.
55.
56.
57.
Moudafi, A., Mainge, P.-E.: Towards viscosity approximations of hierarchical fixed-point problems. Fixed Point Theory Appl. 2006, Article ID 95453 (2006)
58.
Nagurney, A.: Network Economics: A Variational Inequality Approach. Academic Publishers, Dordrecht (1993)CrossRefMATH
59.
Nakajo, K., Takahashi, W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372–379 (2003)CrossRefMATHMathSciNet
60.
O’Hara, J.G., Pillay, P., Xu, H.K.: Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces. Nonlinear Anal. 54, 1417–1426 (2003)CrossRefMATHMathSciNet
61.
Osilike, M.O., Udomene, A.: Demiclosedness principle and convergence theorems for strictly pseudo-contractive mappings of Browder-Petryshyn type. J. Math. Anal. Appl. 256, 431–445 (2001)CrossRefMATHMathSciNet
62.
Outrata, J., Kočvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, Dordrecht (1998)CrossRefMATH
63.
Patriksson, M.: Nonlinear Programming and Variational Inequality Problems: A Unified Approach. Kluwer Academic Publishers, Dordrecht (1999)CrossRefMATH
64.
65.
Sezan, M.I.: An overview of convex projections theory and its application to image recovery problems. Ultramicroscopy 40, 55–67 (1992)CrossRef
66.
Shioji, N., Takahashi, W.: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 125, 3641–3645 (1997)CrossRefMATHMathSciNet
67.
Slavakis, K., Yamada, I.: Robust wideband beamforming by the hybrid steepest descent method. IEEE Trans. Signal Process 55, 4511–4522 (2007)CrossRefMathSciNet
68.
Smith, M.J.: The existence, uniqueness and stability of traffic equilibria. Transp. Res. 13B, 295–304 (1979)
69.
Stampacchia, G.: Formes bilineaires coercivitives sur les ensembles convexes. C.R. Math. Acad. Sci. Paris 258, 4413–4416 (1964)MATHMathSciNet
70.
Stampacchia, G.: Variational inequalities. In: Ghizzetti, A. (ed.) Theory and Applications of Monotone Operators. Proceedings of NATO Advanced Study Institute, Venice, Oderisi, Gubbio (1968)
71.
Takahashi, W., Takeuchi, Y., Kubota, R.: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 341, 276–286 (2008)CrossRefMATHMathSciNet
72.
Udomene, A.: Path convergence, approximation of fixed points and variational solutions of Lipschitz pseudo-contractions in Banach spaces. Nonlinear Anal. 67, 2403–2414 (2007)CrossRefMATHMathSciNet
73.
Wairojjana, N., Jitpeera, T., Kumam, P.: The hybrid steepest descent method for solving variational inequality over triple hierarchical problems. J. Inequal. Appl. 2012, Article ID 280 (2012)
74.
Wu, D.P., Chang, S.S., Yuan, G.X.: Approximation of common fixed points for a family of finite nonexpansive mappings in Banach spaces. Nonlinear Anal. 63, 987–999 (2005)CrossRefMATHMathSciNet
75.
Xu, H.K.: Iterative algorithm for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)CrossRefMATH
76.
77.
Xu, H.K.: Strong convergence of an iterative method for nonexpansive mappings and accretive operators. J. Math. Anal. Appl. 314, 631–643 (2006)CrossRefMATHMathSciNet
78.
Xu, H.K.: Viscosity methods for hierarchical fixed point approach to variational inequalities. Taiwan. J. Math. 14, 463–478 (2010)MATH
79.
Xu, H.K., Kim, T.H.: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory Appl. 119, 185–201 (2003)CrossRefMATHMathSciNet
80.
Yamada, I.: The hybrid steepest descent method for the variational inequality problems over the intersection of fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, pp. 473–504. Elsevier, New York (2001)
81.
Yamada, I., Ogura, N.: Hybrid steepest descent method for the variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings. Numer. Funct. Anal. Optim. 25, 619–655 (2004)CrossRefMATHMathSciNet
82.
Yao, Y., Chen, R., Xu, H.-K.: Schemes for finding minimum-norm solutions of variational inequalities. Nonlinear Anal. 72, 3447–3456 (2010)
83.
Yao, Y., Liou, Y.-C.: Weak and strong convergence of Krasnoselski–Mann iteration for hierarchical fixed point problems. Inverse Prob. 24, Article ID 015015 (2008)
84.
Yao, Y., Liou, Y.-C.: An implicit extragradient method for hierarchical variational inequalities. Fixed Point Theory Appl. 2011, Article ID 697248 (2011)
85.
Yao, Y., Liou, Y.C., Chen, R.: Strong convergence of an iterative algorithm for pseudo-contractive mapping in Banach spaces. Nonlinear Anal. 67, 3311–3317 (2007)CrossRefMATHMathSciNet
86.
87.
Yao, Y., Yao, J.C.: On modified iterative method for nonexpansive mappings and monotone mappings. Appl. Math. Comput. 186, 1551–1558 (2007)CrossRefMATHMathSciNet
88.
Zeng, L.C.: A note on approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 226, 245–250 (1998)CrossRefMATHMathSciNet
89.
Zeng, L.C., Wong, M.M., Yao, J.C.: Strong convergence of relaxed hybrid steepestdescent methods for triple hierarchical constrained optimization. Fixed Point Theory Appl. 2012, Article ID 29 (2012)
90.
Zeng, L.C., Wong, N.C., Yao, J.C.: Strong convergence theorems for strictly pseudo-contractive mappings of Browder-Petryshyn type. Taiwan. J. Math. 10, 837–849 (2006)MATHMathSciNet
91.
Zeng, L.C., Wong, N.C., Yao, J.C.: On the convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities. J. Optim. Theory Appl. 132, 51–69 (2007)CrossRefMATHMathSciNet
92.
Zeng, L.C., Wen, C.-F., Yao, J.C.: An implicit hierarchical fixed-point approach to general variational inequalities in Hilbert spaces. Fixed Point Theory Appl. 2011, Article ID 748918 (2007)
93.
Zeng, L.C., Yao, J.C.: Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings. Nonlinear Anal. 64, 2507–2515 (2006)CrossRefMATHMathSciNet
94.
Zhou, H.Y.: Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 343, 546–556 (2008)CrossRefMATHMathSciNet
95.
Zhou, H.Y.: Convergence theorems of common fixed points for a finite family of Lipschitz pseudo-contractions in Banach spaces. Nonlinear Anal. 68, 2977–2983 (2008)CrossRefMATHMathSciNet
Metadaten
Titel
Triple Hierarchical Variational Inequalities
verfasst von
Qamrul Hasan Ansari
Lu-Chuan Ceng
Himanshu Gupta
Copyright-Jahr
2014
Verlag
Springer India
Buch
Nonlinear Analysis

Print ISBN: 978-81-322-1882-1

Electronic ISBN: 978-81-322-1883-8

Copyright-Jahr: 2014

DOI
https://doi.org/10.1007/978-81-322-1883-8_8