Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 203467, 6 pages
http://dx.doi.org/10.1155/2014/203467
Research Article

A Note on Optimality Conditions for DC Programs Involving Composite Functions

1College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China
2College of Automation, Chongqing University, Chongqing 400030, China
3School of Management, Southwest University of Political Science and Law, Chongqing 401120, China

Received 23 April 2014; Accepted 22 May 2014; Published 29 May 2014

Academic Editor: Chong Li

Copyright © 2014 Xiang-Kai Sun and Hong-Yong Fu. 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

  1. R. Horst and N. V. Thoai, “DC programming: overview,” Journal of Optimization Theory and Applications, vol. 103, no. 1, pp. 1–43, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  2. M. Laghdir, “Optimality conditions and Toland's duality for a nonconvex minimization problem,” Matematichki Vesnik, vol. 55, no. 1-2, pp. 21–30, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. R. I. Boţ, S.-M. Grad, and G. Wanka, “Generalized Moreau-Rockafellar results for composed convex functions,” Optimization, vol. 58, no. 7, pp. 917–933, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. D. H. Fang, C. Li, and K. F. Ng, “Constraint qualifications for extended Farkas's lemmas and Lagrangian dualities in convex infinite programming,” SIAM Journal on Optimization, vol. 20, no. 3, pp. 1311–1332, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. R. I. Boţ, Conjugate Duality in Convex Optimization, Springer, Berlin, Germany, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  6. N. Dinh, T. T. A. Nghia, and G. Vallet, “A closedness condition and its applications to DC programs with convex constraints,” Optimization, vol. 59, no. 3-4, pp. 541–560, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. D. H. Fang, C. Li, and X. Q. Yang, “Stable and total Fenchel duality for DC optimization problems in locally convex spaces,” SIAM Journal on Optimization, vol. 21, no. 3, pp. 730–760, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. D. H. Fang, C. Li, and X. Q. Yang, “Asymptotic closure condition and Fenchel duality for DC optimization problems in locally convex spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 8, pp. 3672–3681, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. D. H. Fang, G. M. Lee, C. Li, and J. C. Yao, “Extended Farkas's lemmas and strong Lagrange dualities for DC infinite programming,” Journal of Nonlinear and Convex Analysis, vol. 14, no. 4, pp. 747–767, 2013. View at Google Scholar · View at MathSciNet
  10. X. K. Sun and S. J. Li, “Duality and Farkas-type results for extended Ky Fan inequalities with DC functions,” Optimization Letters, vol. 7, no. 3, pp. 499–510, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. X.-K. Sun, S.-J. Li, and D. Zhao, “Duality and Farkas-type results for DC infinite programming with inequality constraints,” Taiwanese Journal of Mathematics, vol. 17, no. 4, pp. 1227–1244, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. X.-K. Sun, “Regularity conditions characterizing Fenchel-Lagrange duality and Farkas-type results in DC infinite programming,” Journal of Mathematical Analysis and Applications, vol. 414, no. 2, pp. 590–611, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  13. L. Thibault, “Sequential convex subdifferential calculus and sequential Lagrange multipliers,” SIAM Journal on Control and Optimization, vol. 35, no. 4, pp. 1434–1444, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. V. Jeyakumar, G. M. Lee, and N. Dinh, “New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs,” SIAM Journal on Optimization, vol. 14, no. 2, pp. 534–547, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. V. Jeyakumar, Z. Y. Wu, G. M. Lee, and N. Dinh, “Liberating the subgradient optimality conditions from constraint qualifications,” Journal of Global Optimization, vol. 36, no. 1, pp. 127–137, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. R. I. Boţ, E. R. Csetnek, and G. Wanka, “Sequential optimality conditions for composed convex optimization problems,” Journal of Mathematical Analysis and Applications, vol. 342, no. 2, pp. 1015–1025, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. F. S. Bai, Z. Y. Wu, and D. L. Zhu, “Sequential Lagrange multiplier condition for ϵ-optimal solution in convex programming,” Optimization, vol. 57, no. 5, pp. 669–680, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  18. R. I. Boţ, I. B. Hodrea, and G. Wanka, “ϵ-optimality conditions for composed convex optimization problems,” Journal of Approximation Theory, vol. 153, no. 1, pp. 108–121, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. N. Dinh, B. S. Mordukhovich, and T. T. A. Nghia, “Qualification and optimality conditions for DC programs with infinite constraints,” Acta Mathematica Vietnamica, vol. 34, no. 1, pp. 123–153, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. N. Dinh, J. J. Strodiot, and V. H. Nguyen, “Duality and optimality conditions for generalized equilibrium problems involving DC functions,” Journal of Global Optimization, vol. 48, no. 2, pp. 183–208, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. G. S. Kim and G. M. Lee, “On ϵ-optimality theorems for convex vector optimization problems,” Journal of Nonlinear and Convex Analysis, vol. 12, no. 3, pp. 473–482, 2011. View at Google Scholar · View at MathSciNet
  22. G. M. Lee, G. S. Kim, and N. Dinh, “Optimality conditions for approximate solutions of convex semi-infinite vector optimization problems,” in Recent Developments in Vector Optimization, Vector Optimization, Q. H. Ansari and J. C. Yao, Eds., vol. 1, pp. 275–295, Springer, Berlin, Germany, 2012. View at Google Scholar
  23. X. L. Guo, S. J. Li, and K. L. Teo, “Subdifferential and optimality conditions for the difference of set-valued mappings,” Positivity, vol. 16, no. 2, pp. 321–337, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. X. L. Guo and S. J. Li, “Optimality conditions for vector optimization problems with difference of convex maps,” Journal of Optimization Theory and Applications, 2013. View at Publisher · View at Google Scholar
  25. D. H. Fang and X. P. Zhao, “Local and global optimality conditions for DC infinite optimization problems,” Taiwanese Journal of Mathematics, 2013. View at Publisher · View at Google Scholar
  26. J. B. Hiriart-Urruty, “From convex optimization to nonconvex optimization necessary and sufficient conditions for global optimality,” in From Convexity to Nonconvexity, R. P. Gilbert, P. D. Panagiotopoulos, and P. M. Pardalos, Eds., pp. 219–239, Kluwer, London, UK, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet