- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 792078, 16 pages
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.
- 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.
- P. H. Calamai and J. J. Moré, “Projected gradient methods for linearly constrained problems,” Mathematical Programming, vol. 39, no. 1, pp. 93–116, 1987.
- E. S. Levitin and B. T. Polyak, “Constrained minimization methods,” USSR Computational Mathematics and Mathematical Physics, vol. 6, no. 5, pp. 1–50, 1966.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002.
- 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.
- 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.
- 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.
- H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659–678, 2003.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.