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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 678154, 6 pages
Positive Definiteness of High-Order Subdifferential and High-Order Optimality Conditions in Vector Optimization Problems
1Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
2School of Science, Kunming University of Science and Technology, Kunming, Yunnan 650500, China
Received 15 October 2012; Accepted 26 December 2012
Academic Editor: Gue Lee
Copyright © 2013 He Qinghai and Zhang Binbin. 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.
- D. T. Luc, “Taylor's formula for functions,” SIAM Journal on Optimization, vol. 5, no. 3, pp. 659–669, 1995.
- A. Göpfert, H. Riahi, C. Tammer, and C. Zălinescu, Variational Methods in Partially Ordered Spaces, vol. 17 of CMS Books in Mathematics, Springer, New York, NY, USA, 2003.
- V. V. Gorokhovik, Vypuklye i Negladkie Zadachi Vektornoi Optimizatsii, Nauka i Tekhnika, Minsk, Russia, 1990.
- J. Jahn, Vector Optimization: Theory, Applications and Extensions, Operations Research and Decision Theory, Springer, New York, NY, USA, 2004.
- J. J. Ye and Q. J. Zhu, “Multiobjective optimization problem with variational inequality constraints,” Mathematical Programming, vol. 96, no. 1, pp. 139–160, 2003.
- X. Y. Zheng and X. Q. Yang, “Weak sharp minima for semi-infinite optimization problems with applications,” SIAM Journal on Optimization, vol. 18, no. 2, pp. 573–588, 2007.
- M. Studniarski and D. E. Ward, “Weak sharp minima: characterizations and sufficient conditions,” SIAM Journal on Control and Optimization, vol. 38, no. 1, pp. 219–236, 1999.
- M. I. Henig, “Existence and characterization of efficient decisions with respect to cones,” Mathematical Programming, vol. 23, no. 1, pp. 111–116, 1982.
- G. Isac, “Pareto optimization in infinite-dimensional spaces: the importance of nuclear cones,” Journal of Mathematical Analysis and Applications, vol. 182, no. 2, pp. 393–404, 1994.
- M. Minami, “Weak Pareto-optimal necessary conditions in a nondifferentiable multiobjective program on a Banach space,” Journal of Optimization Theory and Applications, vol. 41, no. 3, pp. 451–461, 1983.
- A. Zaffaroni, “Degrees of efficiency and degrees of minimality,” SIAM Journal on Control and Optimization, vol. 42, no. 3, pp. 1071–1086, 2003.
- X. Y. Zheng, X. M. Yang, and K. L. Teo, “Sharp minima for multiobjective optimization in Banach spaces,” Set-Valued Analysis, vol. 14, no. 4, pp. 327–345, 2006.
- X. Y. Zheng and K. F. Ng, “The Fermat rule for multifunctions on Banach spaces,” Mathematical Programming, vol. 104, no. 1, pp. 69–90, 2005.
- G. Y. Chen, X. X. Huang, and X. Q. Yang, Vector Optimization: Set-Valued and Variational Analysis, Springer, New York, NY, USA, 2005.
- J. V. Burke and S. Deng, “Weak sharp minima revisited. II. Application to linear regularity and error bounds,” Mathematical Programming, vol. 104, no. 2-3, pp. 235–261, 2005.
- R. Ellaia and A. Hassouni, “Characterization of nonsmooth functions through their generalized gradients,” Optimization, vol. 22, no. 3, pp. 401–416, 1991.
- X. Y. Zheng and X. Q. Yang, “Conic positive definiteness and sharp minima of fractional orders in vector optimization problems,” Journal of Mathematical Analysis and Applications, vol. 391, no. 2, pp. 619–629, 2012.
- F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, NY, USA, 1983.
- S. Winfried, Nonsmooth Analysis, Springer, Berlin, Germany, 2007.
- J. B. Hiriart-Urruty, J. J. Strodiot, and V. H. Nguyen, “Generalized Hessian matrix and second-order optimality conditions for problems with data,” Applied Mathematics and Optimization, vol. 11, no. 1, pp. 43–56, 1984.