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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 697474, 7 pages
An Approximate Quasi-Newton Bundle-Type Method for Nonsmooth Optimization
1School of Mathematics, Liaoning Normal University, Dalian 116029, China
2School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
Received 22 January 2013; Revised 31 March 2013; Accepted 1 April 2013
Academic Editor: Gue Lee
Copyright © 2013 Jie Shen 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.
- M. Fukushima and L. Qi, “A globally and superlinearly convergent algorithm for nonsmooth convex minimization,” SIAM Journal on Optimization, vol. 6, no. 4, pp. 1106–1120, 1996.
- A. I. Rauf and M. Fukushima, “Globally convergent BFGS method for nonsmooth convex optimization,” Journal of Optimization Theory and Applications, vol. 104, no. 3, pp. 539–558, 2000.
- L. Qi and X. Chen, “A preconditioning proximal Newton method for nondifferentiable convex optimization,” Mathematical Programming, vol. 76, no. 3, pp. 411–429, 1997.
- Y. R. He, “Minimizing and stationary sequences of convex constrained minimization problems,” Journal of Optimization Theory and Applications, vol. 111, no. 1, pp. 137–153, 2001.
- R. Mifflin and C. Sagastizábal, “A VU-proximal point algorithm for minimization,” in Numerical Optimization, Universitext, Springer, Berlin, Germany, 2002.
- C. Lemaréchal, F. Oustry, and C. Sagastizábal, “The -Lagrangian of a convex function,” Transactions of the American Mathematical Society, vol. 352, no. 2, pp. 711–729, 2000.
- R. Mifflin, D. Sun, and L. Qi, “Quasi-Newton bundle-type methods for nondifferentiable convex optimization,” SIAM Journal on Optimization, vol. 8, no. 2, pp. 583–603, 1998.
- J. B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms, Springer, Berlin, Germany, 1993.
- X. Chen and M. Fukushima, “Proximal quasi-newton methods for nondifferentiable convex optimization,” Applied Mathematics Report 95/32, School of Mathematics, The University of New South Wales, Sydney, Australia, 1995.
- R. Mifflin, “A quasi-second-order proximal bundle algorithm,” Mathematical Programming, vol. 73, no. 1, pp. 51–72, 1996.
- K. C. Kiwiel, “Approximations in proximal bundle methods and decomposition of convex programs,” Journal of Optimization Theory and Applications, vol. 84, no. 3, pp. 529–548, 1995.
- M. Hintermüller, “A proximal bundle method based on approximate subgradients,” Computational Optimization and Applications, vol. 20, no. 3, pp. 245–266, 2001.
- R. Gabasov and F. M. Kirilova, Methods of Linear Programming, Part 3, SpecialProblems, Izdatel'Stov BGU, Minsk, Belarus, 1980 (Russian).
- M. V. Solodov, “On approximations with finite precision in bundle methods for nonsmooth optimization,” Journal of Optimization Theory and Applications, vol. 119, no. 1, pp. 151–165, 2003.
- K. C. Kiwiel, “An algorithm for nonsmooth convex minimization with errors,” Mathematics of Computation, vol. 45, no. 171, pp. 173–180, 1985.
- P. Wolfe, “A method of conjugate subgradients for minimizing nondifferentiable functions,” Mathematical Programming Study, vol. 3, pp. 145–173, 1975.
- C. Lemaréchal, “An extension of davidon methods to non differentiable problems,” Mathematical Programming Study, vol. 3, pp. 95–109, 1975.
- K. C. Kiwiel, A Variable Metric Method of Centres for Nonsmooth Minimization, International Institute for Applied Systems Analysis, Laxemnburg, Austria, 1981.
- K. C. Kiwiel, Efficient algorithms for nonsmooth optimization and their applications [Ph.D. thesis], Department of Electronics, Technical University of Warsaw, Warsaw, Poland, 1982.
- R. Mifflin, “A modification and extension of Lemarechal's algorithm for nonsmooth minimization,” Mathematical Programming Study, vol. 17, pp. 77–90, 1982.
- M. Fukushima, “A descent algorithm for nonsmooth convex optimization,” Mathematical Programming, vol. 30, no. 2, pp. 163–175, 1984.
- L. Q. Qi and R. S. Womersley, “An SQP algorithm for extended linear-quadratic problems in stochastic programming,” Annals of Operations Research, vol. 56, pp. 251–285, 1995.