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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 792078, 16 pages
doi:10.1155/2012/792078
Research Article
Variant Gradient Projection Methods for the Minimization Problems
1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China
2Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan
3Center for General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan
Received 3 May 2012; Accepted 6 June 2012
Academic Editor: Jen-Chin Yao
Copyright © 2012 Yonghong Yao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Linked References
- E. M. Gafni and D. P. Bertsekas, “Two-metric projection methods for constrained optimization,” SIAM Journal on Control and Optimization, vol. 22, no. 6, pp. 936–964, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- P. H. Calamai and J. J. Moré, “Projected gradient methods for linearly constrained problems,” Mathematical Programming, vol. 39, no. 1, pp. 93–116, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- E. S. Levitin and B. T. Polyak, “Constrained minimization methods,” USSR Computational Mathematics and Mathematical Physics, vol. 6, no. 5, pp. 1–50, 1966. View at Scopus
- B. T. Polyak, Introduction to Optimization, Optimization Software, New York, NY, USA, 1987.
- A. Ruszczyński, Nonlinear Optimization, Princeton University Press, Princeton, NJ, USA, 2006.
- C. Wang and N. Xiu, “Convergence of the gradient projection method for generalized convex minimization,” Computational Optimization and Applications, vol. 16, no. 2, pp. 111–120, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- N. Xiu, C. Wang, and J. Zhang, “Convergence properties of projection and contraction methods for variational inequality problems,” Applied Mathematics and Optimization, vol. 43, no. 2, pp. 147–168, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- N. Xiu, C. Wang, and L. Kong, “A note on the gradient projection method with exact stepsize rule,” Journal of Computational Mathematics, vol. 25, no. 2, pp. 221–230, 2007. View at Zentralblatt MATH
- M. Su and H. K. Xu, “Remarks on the gradient-projection algorithm,” Journal of Nonlinear Analysis and Optimization, vol. 1, pp. 35–43, 2010.
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- Y. Censor, T. Elfving, N. Kopf, and T. Bortfeld, “The multiple-sets split feasibility problem and its applications for inverse problems,” Inverse Problems, vol. 21, no. 6, pp. 2071–2084, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- Y. Censor, T. Bortfeld, B. Martin, and A. Trofimov, “A unified approach for inversion problems in intensity-modulated radiation therapy,” Physics in Medicine and Biology, vol. 51, no. 10, pp. 2353–2365, 2006. View at Publisher · View at Google Scholar · View at Scopus
- H.-K. Xu, “A variable Krasnosel'skii–Mann algorithm and the multiple-set split feasibility problem,” Inverse Problems, vol. 22, no. 6, pp. 2021–2034, 2006. View at Publisher · View at Google Scholar
- H.-K. Xu, “Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces,” Inverse Problems, vol. 26, no. 10, Article ID 105018, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- G. Lopez, V. Martin, and H.-K. Xu, “Perturbation techniques for nonexpansive mappings with applications,” Nonlinear Analysis: Real World Applications, vol. 10, no. 4, pp. 2369–2383, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- G. Lopez, V. Martin, and H. K. Xu, “Iterative algorithms for the multiple-sets split feasibility problem,” in Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems, Y. Censor, M. Jiang, and G. Wang, Eds., pp. 243–279, Medical Physics Publishing, Madison, Wis, USA, 2009.
- H.-K. Xu, “Averaged mappings and the gradient-projection algorithm,” Journal of Optimization Theory and Applications, vol. 150, no. 2, pp. 360–378, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- M. V. Solodov and B. F. Svaiter, “A new projection method for variational inequality problems,” SIAM Journal on Control and Optimization, vol. 37, no. 3, pp. 765–776, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990. View at Publisher · View at Google Scholar
- C. Martinez-Yanes and H.-K. Xu, “Strong convergence of the CQ method for fixed point iteration processes,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 11, pp. 2400–2411, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- T. Suzuki, “Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces,” Fixed Point Theory and Applications, vol. 2005, no. 1, pp. 103–123, 2005. View at Zentralblatt MATH
- S. Reich and H.-K. Xu, “An iterative approach to a constrained least squares problem,” Abstract and Applied Analysis, no. 8, pp. 503–512, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- A. Sabharwal and L. C. Potter, “Convexly constrained linear inverse problems: iterative least-squares and regularization,” IEEE Transactions on Signal Processing, vol. 46, no. 9, pp. 2345–2352, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659–678, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- F. Cianciaruso, G. Marino, L. Muglia, and Y. Yao, “A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem,” Fixed Point Theory and Applications, vol. 2010, Article ID 383740, 19 pages, 2010. View at Zentralblatt MATH
- F. Cianciaruso, G. Marino, L. Muglia, and Y. Yao, “On a two-step algorithm for hierarchical fixed point problems and variational inequalities,” Journal of Inequalities and Applications, vol. 2009, Article ID 208692, 13 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- Y. Yao, Y. J. Cho, and Y.-C. Liou, “Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems,” European Journal of Operational Research, vol. 212, no. 2, pp. 242–250, 2011. View at Publisher · View at Google Scholar
- Y. Yao, Y.-C. Liou, and S. M. Kang, “Two-step projection methods for a system of variational inequality problems in Banach spaces,” Journal of Global Optimization. In press. View at Publisher · View at Google Scholar
- Y. Yao, R. Chen, and Y.-C. Liou, “A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem,” Mathematical Mathematical & Computer Modelling, vol. 55, pp. 1506–1515, 2012. View at Publisher · View at Google Scholar
- H. H. Bauschke and J. M. Borwein, “On projection algorithms for solving convex feasibility problems,” SIAM Review, vol. 38, no. 3, pp. 367–426, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- K. C. Kiwiel and B. Łopuch, “Surrogate projection methods for finding fixed points of firmly nonexpansive mappings,” SIAM Journal on Optimization, vol. 7, no. 4, pp. 1084–1102, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- K. C. Kiwiel, “The efficiency of subgradient projection methods for convex optimization. I. General level methods,” SIAM Journal on Control and Optimization, vol. 34, no. 2, pp. 660–676, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
- K. C. Kiwiel, “The efficiency of subgradient projection methods for convex optimization. II. Implementations and extensions,” SIAM Journal on Control and Optimization, vol. 34, no. 2, pp. 677–697, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH