Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2017, Article ID 1564642, 15 pages
https://doi.org/10.1155/2017/1564642
Research Article

A Quasi-Monte-Carlo-Based Feasible Sequential System of Linear Equations Method for Stochastic Programs with Recourse

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Changyin Zhou; moc.361@321ycuohz

Received 5 April 2017; Revised 14 July 2017; Accepted 24 July 2017; Published 24 August 2017

Academic Editor: Huanqing Wang

Copyright © 2017 Changyin Zhou 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.

Linked References

  1. J. R. Birge and F. Louveaux, Introduction to Stochastic Programming, Springer, New York, NY, USA, 1997. View at MathSciNet
  2. S. W. Wallace and W. T. Ziemba, “Applications of stochastic programming,” in Proceedings of the MPS/SIAM Series on Optimization, vol. 5, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2005.
  3. X. Liu, Y. Li, and W. Zhang, “Stochastic linear quadratic optimal control with constraint for discrete-time systems,” Applied Mathematics and Computation, vol. 228, pp. 264–270, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. G. Li and W. Zhang, “Study on indefinite stochastic linear quadratic optimal control with inequality constraint,” Journal of Applied Mathematics, vol. 2013, Article ID 805829, 5004 pages, 2013. View at Publisher · View at Google Scholar · View at Scopus
  5. L. Liu and X. Meng, “Optimal harvesting control and dynamics of two-species stochastic model with delays,” Advances in Difference Equations, vol. 2017, 18 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  6. G. Li and M. Chen, “Infinite horizon linear quadratic optimal control for stochastic difference time-delay systems,” Advances in Difference Equations, vol. 2015, article 14, 2017. View at Google Scholar
  7. H. Yang, “Study on stochastic linear quadratic optimal control with quadratic and mixed terminal state constraints,” Journal of Applied Mathematics, vol. 2013, Article ID 674327, 11 pages, 2013. View at Publisher · View at Google Scholar · View at Scopus
  8. A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro, “Robust stochastic approximation approach to stochastic programming,” SIAM Journal on Optimization, vol. 19, no. 4, pp. 1574–1609, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  9. J. Linderoth, A. Shapiro, and S. Wright, “The empirical behavior of sampling methods for stochastic programming,” Annals of Operations Research, vol. 142, no. 1, pp. 215–241, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  10. J. Yu, M. Li, Y. Wang, and G. He, “A decomposition method for large-scale box constrained optimization,” Applied Mathematics and Computation, vol. 231, no. 12, pp. 9–15, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  11. G. Bayraksan and D. P. Morton, “A sequential sampling procedure for stochastic programming,” Operations Research, vol. 59, no. 4, pp. 898–913, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. T. Ekin, N. G. Polson, and R. Soyer, “Augmented Markov chain MONte Carlo simulation for two-stage stochastic programs with recourse,” Decision Analysis, vol. 11, no. 4, pp. 250–264, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. R. T. Rockafellar and R. J. Wets, “A dual solution procedure for quadratic stochastic programs with simple recourse,” in Numerical methods, vol. 1005 of Lecture Notes in Math, pp. 252–265, Springer, Berlin, Germany, 1983. View at Publisher · View at Google Scholar · View at MathSciNet
  14. R. T. Rockafellar and R. J. Wets, “A Lagrangian finite generation technique for solving linear-quadratic problems in stochastic programming,” Mathematical Programming Study, no. 28, pp. 63–93, 1986. View at Google Scholar · View at MathSciNet · View at Scopus
  15. 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. X. J. Chen, L. Q. Qi, and R. S. Womersley, “Newton's method for quadratic stochastic programs with recourse,” Journal of Computational and Applied Mathematics, vol. 60, no. 1-2, pp. 29–46, 1995. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. J. Birge, X. Chen, and L. Qi, “A stochastic Newton method for stochastic quadratic programs with recourse,” https://www.researchgate.net/publication/2255017.
  18. S. Joe and I. H. Sloan, “Imbedded lattice rules for multidimensional integration,” SIAM Journal on Numerical Analysis, vol. 29, no. 4, pp. 1119–1135, 1992. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. S. Joe and I. H. Sloan, “Implementation of a Lattice Method for Numerical Multiple Integration,” ACM Transactions on Mathematical Software (TOMS), vol. 19, no. 4, pp. 523–545, 1993. View at Publisher · View at Google Scholar · View at Scopus
  20. H. Niederreiter, “Random number generation and quasi-Monte Carlo methods,” Journal of the American Statistical Association, vol. 88, no. 89, pp. 147–153, 1992. View at Google Scholar
  21. J. R. Birge, “Quasi-Monte Carlo approaches to option pricing,” American Anthropologist, vol. 81, no. 2, pp. 87–388, 1995. View at Google Scholar
  22. E. R. Panier, A. L. Tits, and J. N. Herskovits, “A QP-free, globally convergent, locally superlinearly convergent algorithm for inequality constrained optimization,” SIAM Journal on Control and Optimization, vol. 26, no. 4, pp. 788–811, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  23. L. Sun, G. He, Y. Wang, and C. Zhou, “An accurate active set Newton algorithm for large scale bound constrained optimization,” Applications of Mathematics, vol. 56, no. 3, pp. 297–314, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. Y. Li, T. Tan, and X. Li, “A log-exponential smoothing method for mathematical programs with complementarity constraints,” Applied Mathematics and Computation, vol. 218, no. 10, pp. 5900–5909, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. H.-D. Qi and L. Qi, “A new {QP}-free, globally convergent, locally superlinearly convergent algorithm for inequality constrained optimization,” SIAM Journal on Optimization, vol. 11, no. 1, pp. 113–132, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. Z. Y. Gao, G. P. He, and F. Wu, “Sequential systems of linear equations algorithm for nonlinear optimization problems with general constraints,” Journal of Optimization Theory and Applications, vol. 147, no. 3, pp. 211–226, 2002. View at Google Scholar
  27. Y.-F. Yang, D.-H. Li, and L. Qi, “A feasible sequential linear equation method for inequality constrained optimization,” SIAM Journal on Optimization, vol. 13, no. 4, pp. 1222–1244, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. F. Facchinei, A. Fischer, and C. Kanzow, “On the accurate identification of active constraints,” SIAM Journal on Optimization, vol. 9, no. 1, pp. 14–32, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  29. M. J. D. Powell, “A fast algorithm for nonlinearly constrained optimization calculations, in Numerical Analysis,” Lecture Notes in Mathematics, vol. 630, pp. 144–157, 1978. View at Google Scholar
  30. S. M. Robinson, “Strongly regular generalized equations,” Mathematics of Operations Research, vol. 5, no. 1, pp. 43–62, 1980. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. C. Han, F. Zheng, T. Guo, and G. He, “Parallel algorithms for large-scale linearly constrained minimization problem,” Acta Mathematicae Applicatae Sinica, vol. 30, no. 3, pp. 707–720, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  32. C. Han, T. Feng, G. He, and T. Guo, “Parallel variable distribution algorithm for constrained optimization with nonmonotone technique,” Journal of Applied Mathematics, vol. 2013, Article ID 295147, 420 pages, 2013. View at Publisher · View at Google Scholar · View at Scopus
  33. F. Zheng, C. Han, and Y. Wang, “Parallel {SSLE} algorithm for large scale constrained optimization,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5377–5384, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus