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An Iterative Approach to Quadratic Optimization

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  • H.K. Xu

    (University of Durban-Westville)

Abstract

Assume that C 1, . . . , C N are N closed convex subsets of a real Hilbert space H having a nonempty intersection C. Assume also that each C i is the fixed point set of a nonexpansive mapping T i of H. We devise an iterative algorithm which generates a sequence (x n ) from an arbitrary initial x 0∈H. The sequence (xn) is shown to converge in norm to the unique solution of the quadratic minimization problem min x∈C (1/2)〈Ax, x〉−〈x, u〉, where A is a bounded linear strongly positive operator on H and u is a given point in H. Quadratic–quadratic minimization problems are also discussed.

Suggested Citation

  • H.K. Xu, 2003. "An Iterative Approach to Quadratic Optimization," Journal of Optimization Theory and Applications, Springer, vol. 116(3), pages 659-678, March.
    • Handle: RePEc:spr:joptap:v:116:y:2003:i:3:d:10.1023_a:1023073621589
    DOI: 10.1023/A:1023073621589
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    References listed on IDEAS

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    1. H. K. Xu & T. H. Kim, 2003. "Convergence of Hybrid Steepest-Descent Methods for Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 119(1), pages 185-201, October.
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    Cited by:

    1. Eric U. Ofoedu, 2013. "A General Approximation Scheme for Solutions of Various Problems in Fixed Point Theory," International Journal of Analysis, Hindawi, vol. 2013, pages 1-18, January.
    2. Nguyen Buong & Lam Thuy Duong, 2011. "An Explicit Iterative Algorithm for a Class of Variational Inequalities in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 151(3), pages 513-524, December.
    3. Haiyun Zhou & Peiyuan Wang, 2014. "A Simpler Explicit Iterative Algorithm for a Class of Variational Inequalities in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 716-727, June.
    4. Jeong, Jae Ug, 2016. "Generalized viscosity approximation methods for mixed equilibrium problems and fixed point problems," Applied Mathematics and Computation, Elsevier, vol. 283(C), pages 168-180.
    5. Mohammad Eslamian & Ali Abkar, 2014. "Viscosity iterative scheme for generalized mixed equilibrium problems and nonexpansive semigroups," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(2), pages 554-570, July.
    6. Vahid Darvish, 2016. "Strong convergence theorem for a system of generalized mixed equilibrium problems and finite family of Bregman nonexpansive mappings in Banach spaces," OPSEARCH, Springer;Operational Research Society of India, vol. 53(3), pages 584-603, September.
    7. ChangAn Tian & Yisheng Song, 2013. "Strong convergence of a regularization method for Rockafellar’s proximal point algorithm," Journal of Global Optimization, Springer, vol. 55(4), pages 831-837, April.
    8. Uthai Kamraksa & Rabian Wangkeeree, 2011. "Generalized equilibrium problems and fixed point problems for nonexpansive semigroups in Hilbert spaces," Journal of Global Optimization, Springer, vol. 51(4), pages 689-714, December.
    9. Phayap Katchang & Poom Kumam, 2013. "An iterative algorithm for common fixed points for nonexpansive semigroups and strictly pseudo-contractive mappings with optimization problems," Journal of Global Optimization, Springer, vol. 56(4), pages 1563-1589, August.
    10. Rattanaporn Wangkeeree & Narin Petrot & Rabian Wangkeeree, 2011. "The general iterative methods for nonexpansive mappings in Banach spaces," Journal of Global Optimization, Springer, vol. 51(1), pages 27-46, September.
    11. Rattanaporn Wangkeeree & Rabian Wangkeeree, 2013. "The general iterative methods for nonexpansive semigroups in Banach spaces," Journal of Global Optimization, Springer, vol. 55(2), pages 417-436, February.
    12. F. Cianciaruso & G. Marino & L. Muglia, 2010. "Iterative Methods for Equilibrium and Fixed Point Problems for Nonexpansive Semigroups in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 491-509, August.
    13. H. K. Xu & T. H. Kim, 2003. "Convergence of Hybrid Steepest-Descent Methods for Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 119(1), pages 185-201, October.
    14. Atid Kangtunyakarn, 2013. "A new iterative scheme for fixed point problems of infinite family of κ i -pseudo contractive mappings, equilibrium problem, variational inequality problems," Journal of Global Optimization, Springer, vol. 56(4), pages 1543-1562, August.
    15. Yonghong Yao & Yeong-Cheng Liou & Shin Kang, 2010. "Minimization of equilibrium problems, variational inequality problems and fixed point problems," Journal of Global Optimization, Springer, vol. 48(4), pages 643-656, December.
    16. Sitthithakerngkiet, Kanokwan & Deepho, Jitsupa & Kumam, Poom, 2015. "A hybrid viscosity algorithm via modify the hybrid steepest descent method for solving the split variational inclusion in image reconstruction and fixed point problems," Applied Mathematics and Computation, Elsevier, vol. 250(C), pages 986-1001.
    17. Giuseppe Marino & Hong-Kun Xu, 2011. "Explicit Hierarchical Fixed Point Approach to Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 149(1), pages 61-78, April.
    18. Abkar, A. & Tavakkoli, M., 2015. "A new algorithm for two finite families of demicontractive mappings and equilibrium problems," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 491-500.
    19. L. C. Zeng & N. C. Wong & J. C. Yao, 2007. "Convergence Analysis of Modified Hybrid Steepest-Descent Methods with Variable Parameters for Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 132(1), pages 51-69, January.
    20. Scott Kolodziej & Pedro Castro & Ignacio Grossmann, 2013. "Global optimization of bilinear programs with a multiparametric disaggregation technique," Journal of Global Optimization, Springer, vol. 57(4), pages 1039-1063, December.
    21. Wachirapong Jirakitpuwapat & Poom Kumam & Yeol Je Cho & Kanokwan Sitthithakerngkiet, 2019. "A General Algorithm for the Split Common Fixed Point Problem with Its Applications to Signal Processing," Mathematics, MDPI, vol. 7(3), pages 1-20, February.
    22. S. Saeidi & D. S. Kim, 2014. "Combination of the Hybrid Steepest-Descent Method and the Viscosity Approximation," Journal of Optimization Theory and Applications, Springer, vol. 160(3), pages 911-930, March.
    23. Shengquan Weng & Dingping Wu, 2018. "Convergence Theorems of Modified Proximal Algorithms for Asymptotical Quasi-nonexpansive Mappings in CAT(0) Spaces," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 10(2), pages 66-76, April.
    24. Jamil, Muhammad Ahmad & Zubair, Syed M., 2017. "Design and analysis of a forward feed multi-effect mechanical vapor compression desalination system: An exergo-economic approach," Energy, Elsevier, vol. 140(P1), pages 1107-1120.

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