IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v7y2019i5p435-d231913.html
   My bibliography  Save this article

Numerical Method for Solving the Robust Continuous-Time Linear Programming Problems

Author

Listed:
  • Hsien-Chung Wu

    (Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 802, Taiwan)

Abstract

A robust continuous-time linear programming problem is formulated and solved numerically in this paper. The data occurring in the continuous-time linear programming problem are assumed to be uncertain. In this paper, the uncertainty is treated by following the concept of robust optimization, which has been extensively studied recently. We introduce the robust counterpart of the continuous-time linear programming problem. In order to solve this robust counterpart, a discretization problem is formulated and solved to obtain the ? -optimal solution. The important contribution of this paper is to locate the error bound between the optimal solution and ? -optimal solution.

Suggested Citation

  • Hsien-Chung Wu, 2019. "Numerical Method for Solving the Robust Continuous-Time Linear Programming Problems," Mathematics, MDPI, vol. 7(5), pages 1-50, May.
  • Handle: RePEc:gam:jmathe:v:7:y:2019:i:5:p:435-:d:231913
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/7/5/435/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/7/5/435/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Y. Zhang, 2007. "General Robust-Optimization Formulation for Nonlinear Programming," Journal of Optimization Theory and Applications, Springer, vol. 132(1), pages 111-124, January.
    2. A. Ben-Tal & A. Nemirovski, 1998. "Robust Convex Optimization," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 769-805, November.
    3. Ching-Feng Wen & Hsien-Chung Wu, 2011. "Using the Dinkelbach-type algorithm to solve the continuous-time linear fractional programming problems," Journal of Global Optimization, Springer, vol. 49(2), pages 237-263, February.
    4. S. Nobakhtian & M. R. Pouryayevali, 2008. "Duality for Nonsmooth Continuous-Time Problems of Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 136(1), pages 77-85, January.
    5. Lisa Fleischer & Jay Sethuraman, 2005. "Efficient Algorithms for Separated Continuous Linear Programs: The Multicommodity Flow Problem with Holding Costs and Extensions," Mathematics of Operations Research, INFORMS, vol. 30(4), pages 916-938, November.
    6. S. Nobakhtian & M. R. Pouryayevali, 2008. "Optimality Criteria for Nonsmooth Continuous-Time Problems of Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 136(1), pages 69-76, January.
    7. Dimitris Bertsimas & Melvyn Sim, 2004. "The Price of Robustness," Operations Research, INFORMS, vol. 52(1), pages 35-53, February.
    8. George B. Dantzig, 1955. "Linear Programming under Uncertainty," Management Science, INFORMS, vol. 1(3-4), pages 197-206, 04-07.
    9. Xin Chen & Melvyn Sim & Peng Sun, 2007. "A Robust Optimization Perspective on Stochastic Programming," Operations Research, INFORMS, vol. 55(6), pages 1058-1071, December.
    10. Ching-Feng Wen & Hsien-Chung Wu, 2012. "Using the parametric approach to solve the continuous-time linear fractional max–min problems," Journal of Global Optimization, Springer, vol. 54(1), pages 129-153, September.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Hsien-Chung Wu, 2021. "Robust Solutions for Uncertain Continuous-Time Linear Programming Problems with Time-Dependent Matrices," Mathematics, MDPI, vol. 9(8), pages 1-52, April.
    2. Han, Biao & Shang, Chao & Huang, Dexian, 2021. "Multiple kernel learning-aided robust optimization: Learning algorithm, computational tractability, and usage in multi-stage decision-making," European Journal of Operational Research, Elsevier, vol. 292(3), pages 1004-1018.
    3. Wenqing Chen & Melvyn Sim & Jie Sun & Chung-Piaw Teo, 2010. "From CVaR to Uncertainty Set: Implications in Joint Chance-Constrained Optimization," Operations Research, INFORMS, vol. 58(2), pages 470-485, April.
    4. Hamed Mamani & Shima Nassiri & Michael R. Wagner, 2017. "Closed-Form Solutions for Robust Inventory Management," Management Science, INFORMS, vol. 63(5), pages 1625-1643, May.
    5. L. Jeff Hong & Zhiyuan Huang & Henry Lam, 2021. "Learning-Based Robust Optimization: Procedures and Statistical Guarantees," Management Science, INFORMS, vol. 67(6), pages 3447-3467, June.
    6. Fanwen Meng & Jin Qi & Meilin Zhang & James Ang & Singfat Chu & Melvyn Sim, 2015. "A Robust Optimization Model for Managing Elective Admission in a Public Hospital," Operations Research, INFORMS, vol. 63(6), pages 1452-1467, December.
    7. Wenqing Chen & Melvyn Sim, 2009. "Goal-Driven Optimization," Operations Research, INFORMS, vol. 57(2), pages 342-357, April.
    8. Dimitris Bertsimas & Vineet Goyal, 2013. "On the approximability of adjustable robust convex optimization under uncertainty," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(3), pages 323-343, June.
    9. Karthik Natarajan & Dessislava Pachamanova & Melvyn Sim, 2008. "Incorporating Asymmetric Distributional Information in Robust Value-at-Risk Optimization," Management Science, INFORMS, vol. 54(3), pages 573-585, March.
    10. Joel Goh & Melvyn Sim, 2011. "Robust Optimization Made Easy with ROME," Operations Research, INFORMS, vol. 59(4), pages 973-985, August.
    11. Shipra Agrawal & Yichuan Ding & Amin Saberi & Yinyu Ye, 2012. "Price of Correlations in Stochastic Optimization," Operations Research, INFORMS, vol. 60(1), pages 150-162, February.
    12. Gabrel, Virginie & Murat, Cécile & Thiele, Aurélie, 2014. "Recent advances in robust optimization: An overview," European Journal of Operational Research, Elsevier, vol. 235(3), pages 471-483.
    13. Long He & Ho-Yin Mak & Ying Rong & Zuo-Jun Max Shen, 2017. "Service Region Design for Urban Electric Vehicle Sharing Systems," Manufacturing & Service Operations Management, INFORMS, vol. 19(2), pages 309-327, May.
    14. Evers, L. & Glorie, K.M. & van der Ster, S. & Barros, A.I. & Monsuur, H., 2012. "The Orienteering Problem under Uncertainty Stochastic Programming and Robust Optimization compared," Econometric Institute Research Papers EI 2012-21, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    15. Dimitris Bertsimas & Melvyn Sim & Meilin Zhang, 2019. "Adaptive Distributionally Robust Optimization," Management Science, INFORMS, vol. 65(2), pages 604-618, February.
    16. Huan Xu & Constantine Caramanis & Shie Mannor, 2012. "Optimization Under Probabilistic Envelope Constraints," Operations Research, INFORMS, vol. 60(3), pages 682-699, June.
    17. Emilio Carrizosa & Frédéric Messine, 2021. "An interval branch and bound method for global Robust optimization," Journal of Global Optimization, Springer, vol. 80(3), pages 507-522, July.
    18. Ching-Feng Wen & Hsien-Chung Wu, 2011. "Using the Dinkelbach-type algorithm to solve the continuous-time linear fractional programming problems," Journal of Global Optimization, Springer, vol. 49(2), pages 237-263, February.
    19. Elodie Adida & Georgia Perakis, 2010. "Dynamic pricing and inventory control: robust vs. stochastic uncertainty models—a computational study," Annals of Operations Research, Springer, vol. 181(1), pages 125-157, December.
    20. Jia Shu & Miao Song, 2014. "Dynamic Container Deployment: Two-Stage Robust Model, Complexity, and Computational Results," INFORMS Journal on Computing, INFORMS, vol. 26(1), pages 135-149, February.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:7:y:2019:i:5:p:435-:d:231913. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.