Journal of Applied Mathematics

A General Approximation Method for a Kind of Convex Optimization Problems in Hilbert Spaces

Ming Tian and Li-Hua Huang

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The constrained convex minimization problem is to find a point x with the property that x C , and h ( x ) = min    h ( x ) , x C , where C is a nonempty, closed, and convex subset of a real Hilbert space H , h ( x ) is a real-valued convex function, and h ( x ) is not Fréchet differentiable, but lower semicontinuous. In this paper, we discuss an iterative algorithm which is different from traditional gradient-projection algorithms. We firstly construct a bifunction F 1 ( x , y ) defined as F 1 ( x , y ) = h ( y ) h ( x ) . And we ensure the equilibrium problem for F 1 ( x , y ) equivalent to the above optimization problem. Then we use iterative methods for equilibrium problems to study the above optimization problem. Based on Jung’s method (2011), we propose a general approximation method and prove the strong convergence of our algorithm to a solution of the above optimization problem. In addition, we apply the proposed iterative algorithm for finding a solution of the split feasibility problem and establish the strong convergence theorem. The results of this paper extend and improve some existing results.

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J. Appl. Math., Volume 2014 (2014), Article ID 156073, 9 pages.

First available in Project Euclid: 2 March 2015

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Tian, Ming; Huang, Li-Hua. A General Approximation Method for a Kind of Convex Optimization Problems in Hilbert Spaces. J. Appl. Math. 2014 (2014), Article ID 156073, 9 pages. doi:10.1155/2014/156073.

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  • W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Japan, 2000.
  • Y. F. Su and M. Q. Li, “Approximation of solutions of equilibrium problems and application in optimization problems,” Journal of Applied Mathematics and Mechanics, vol. 6, pp. 82–93, 2010.
  • I. Yamada, “The hybrid steepest descent for the variational inequality problems over the intersection of fixed points sets of nonexpansive mapping,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Application, D. Butnariu, Y. Censor, and S. Reich, Eds., pp. 473–504, Elservier, New York, NY, USA, 2001.
  • G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006.
  • M. Tian, “A general iterative algorithm for nonexpansive mappings in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 3, pp. 689–694, 2010.
  • J. S. Jung, “Some results on a general iterative method for $k$-strictly pseudo-contractive mappings,” Fixed Point Theory and Applications, vol. 2011, article 24, 2011.
  • H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam, The Netherlands, 1973.
  • C. Jaiboon and P. Kumam, “A hybrid extragradient viscosity approximation method for solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2009, Article ID 374815, 2009.
  • T. Jitpeera, P. Katchang, and P. Kumam, “A viscosity of Cesàro mean approximation methods for a mixed equilibrium, variational inequalities, and fixed point problems,” Fixed Point Theory and Applications, vol. 2011, Article ID 945051, 2011.
  • D. P. Bertsekas and E. M. Gafni, “Projection methods for variational inequalities with application to the traffic assignment problem,” Mathematical Programming Study, no. 17, pp. 139–159, 1982.
  • D. Han and H. K. Lo, “Solving non-additive traffic assignment problems: a descent method for co-coercive variational inequalities,” European Journal of Operational Research, vol. 159, no. 3, pp. 529–544, 2004.
  • E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.
  • S. D. Flåm and A. S. Antipin, “Equilibrium programming using proximal-like algorithms,” Mathematical Programming, vol. 78, no. 1, pp. 29–41, 1997.
  • H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society. Second Series, vol. 66, no. 1, pp. 240–256, 2002.
  • Y. Censor and T. Elfving, “A multiprojection algorithm using Bregman projections in a product space,” Numerical Algorithms, vol. 8, no. 2–4, pp. 221–239, 1994.
  • G. López, V. Martín-Márquez, F. Wang, and H.-K. Xu, “Solving the split feasibility problem without prior knowledge of matrix norms,” Inverse Problems, vol. 28, no. 8, p. 18, article 085004, 2012.
  • J. Zhao, Y. Zhang, and Q. Yang, “Modified projection methods for the split feasibility problem and the multiple-sets split feasibility problem,” Applied Mathematics and Computation, vol. 219, no. 4, pp. 1644–1653, 2012.
  • H.-K. Xu, “Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces,” Inverse Problems, vol. 26, no. 10, p. 17, article 105018, 2010.
  • C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems, vol. 20, no. 1, pp. 103–120, 2004. \endinput