Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2014 (2014), Article ID 896591, 20 pages
http://dx.doi.org/10.1155/2014/896591
Research Article

Stochastic Separated Continuous Conic Programming: Strong Duality and a Solution Method

Lingnan (University) College, Sun Yat-sen University, Guangzhou, Guangdong 510275, China

Received 3 November 2013; Accepted 29 November 2013; Published 9 January 2014

Academic Editor: Dongdong Ge

Copyright © 2014 Xiaoqing Wang. 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. J. R. Birge and F. Louveaux, Introduction to Stochastic Programming, Springer, New York, NY, USA, 1997. View at MathSciNet
  2. X. Wang, S. Zhang, and D. D. Yao, “Separated continuous conic programming: strong duality and an approximation algorithm,” SIAM Journal on Control and Optimization, vol. 48, no. 4, pp. 2118–2138, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. A. Shapiro, D. Dentcheva, and A. Ruszczyński, Lectures on Stochastic Programming: Modeling and Theory, vol. 9 of MOS/SIAM Series on Optimization, SIAM, Philadelphia, Pa, USA, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  4. A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, MPS/SIAM Series on Optimization, SIAM, Philadelphia, Pa, USA, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  5. R. Bellman, “Bottleneck problems and dynamic programming,” Proceedings of the National Academy of Sciences of the United States of America, vol. 39, pp. 947–951, 1953. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. R. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ, USA, 1957. View at MathSciNet
  7. E. J. Anderson, A continuous model for job-shop scheduling [Ph.D. thesis], University of Cambridge, Cambridge, UK, 1978.
  8. E. J. Anderson, P. Nash, and A. F. Perold, “Some properties of a class of continuous linear programs,” SIAM Journal on Control and Optimization, vol. 21, no. 5, pp. 758–765, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. E. J. Anderson and A. B. Philpott, “A continuous-time network simplex algorithm,” Networks, vol. 19, no. 4, pp. 395–425, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. E. J. Anderson and A. B. Philpott, “On the solutions of a class of continuous linear programs,” SIAM Journal on Control and Optimization, vol. 32, no. 5, pp. 1289–1296, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. E. J. Anderson and P. Nash, Linear Programming in Infinite-Dimensional Spaces: Theory and Applications, John Wiley & Sons, Chichester, UK, 1987. View at MathSciNet
  12. M. C. Pullan, “An algorithm for a class of continuous linear programs,” SIAM Journal on Control and Optimization, vol. 31, no. 6, pp. 1558–1577, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. M. C. Pullan, “Forms of optimal solutions for separated continuous linear programs,” SIAM Journal on Control and Optimization, vol. 33, no. 6, pp. 1952–1977, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. M. C. Pullan, “A duality theory for separated continuous linear programs,” SIAM Journal on Control and Optimization, vol. 34, no. 3, pp. 931–965, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. M. C. Pullan, “Linear optimal control problems with piecewise analytic solutions,” Journal of Mathematical Analysis and Applications, vol. 197, no. 1, pp. 207–226, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. M. C. Pullan, “A study of general dynamic network programs with arc time-delays,” SIAM Journal on Optimization, vol. 7, no. 4, pp. 889–912, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. M. C. Pullan, “Convergence of a general class of algorithms for separated continuous linear programs,” SIAM Journal on Optimization, vol. 10, no. 3, pp. 722–731, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. M. C. Pullan, “An extended algorithm for separated continuous linear programs,” Mathematical Programming, vol. 93, no. 3, pp. 415–451, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. X. Luo and D. Bertsimas, “A new algorithm for state-constrained separated continuous linear programs,” SIAM Journal on Control and Optimization, vol. 37, no. 1, pp. 177–210, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. A. Shapiro, “On duality theory of conic linear problems,” in Semi-Infinite Programming, M. A. Goberna and M. A. Lopez, Eds., vol. 57, chapter 7, pp. 135–165, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. L. Fleischer and J. Sethuraman, “Efficient algorithms for separated continuous linear programs: the multicommodity flow problem with holding costs and extensions,” Mathematics of Operations Research, vol. 30, no. 4, pp. 916–938, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. G. Weiss, “A simplex based algorithm to solve separated continuous linear programs,” Mathematical Programming, vol. 115, no. 1, pp. 151–198, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. M. M. Nasrabadi, M. A. Yaghoobi, and M. Mashinchi, “Separated continuous linear programs with fuzzy valued objective function,” Scientia Iranica, vol. 17, no. 2, pp. 105–118, 2010. View at Google Scholar · View at Scopus
  24. X. Wang, “The duality theory for generalized separated continuous conic programming,” in Proceedings of the 4th International Symposium on Computational Intelligence and Design (ISCID '11), pp. 335–339, Hangzhou, China, October 2011. View at Publisher · View at Google Scholar · View at Scopus
  25. X. Wang, “An approximation algorithm for solving generalized separated continuous conic programming,” Advanced Materials Research, vol. 452-453, pp. 1127–1132, 2012. View at Publisher · View at Google Scholar · View at Scopus
  26. V. Chvátal, Linear Programming, W. H. Freeman and Company, New York, NY, USA, 1983. View at MathSciNet
  27. J. F. Sturm, “Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones,” Optimization Methods and Software, vol. 11-12, no. 1–4, pp. 625–653, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. 2013, http://cvxopt.org/index.html.