IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v201y2012i1p159-16710.1007-s10479-012-1199-x.html
   My bibliography  Save this article

Strong polynomiality of the Gass-Saaty shadow-vertex pivoting rule for controlled random walks

Author

Listed:
  • Guy Even
  • Alexander Zadorojniy

Abstract

We consider the subclass of linear programs that formulate Markov Decision Processes ( mdps). We show that the Simplex algorithm with the Gass-Saaty shadow-vertex pivoting rule is strongly polynomial for a subclass of mdps, called controlled random walks (CRWs); the running time is O(|S| 3 ⋅|U| 2 ), where |S| denotes the number of states and |U| denotes the number of actions per state. This result improves the running time of Zadorojniy et al. (Mathematics of Operations Research 34(4):992–1007, 2009 ) algorithm by a factor of |S|. In particular, the number of iterations needed by the Simplex algorithm for CRWs is linear in the number of states and does not depend on the discount factor. Copyright Springer Science+Business Media, LLC 2012

Suggested Citation

  • Guy Even & Alexander Zadorojniy, 2012. "Strong polynomiality of the Gass-Saaty shadow-vertex pivoting rule for controlled random walks," Annals of Operations Research, Springer, vol. 201(1), pages 159-167, December.
    • Handle: RePEc:spr:annopr:v:201:y:2012:i:1:p:159-167:10.1007/s10479-012-1199-x
    DOI: 10.1007/s10479-012-1199-x
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10479-012-1199-x
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10479-012-1199-x?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Richard F. Serfozo, 1979. "Technical Note—An Equivalence Between Continuous and Discrete Time Markov Decision Processes," Operations Research, INFORMS, vol. 27(3), pages 616-620, June.
    2. M. Yadin & P. Naor, 1967. "On queueing systems with variable service capacities," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 14(1), pages 43-53.
    3. Alan S. Manne, 1960. "Linear Programming and Sequential Decisions," Management Science, INFORMS, vol. 6(3), pages 259-267, April.
    4. Éva Tardos, 1986. "A Strongly Polynomial Algorithm to Solve Combinatorial Linear Programs," Operations Research, INFORMS, vol. 34(2), pages 250-256, April.
    5. Karl-Heinz Borgwardt, 1982. "Some Distribution-Independent Results About the Asymptotic Order of the Average Number of Pivot Steps of the Simplex Method," Mathematics of Operations Research, INFORMS, vol. 7(3), pages 441-462, August.
    6. Cyrus Derman, 1962. "On Sequential Decisions and Markov Chains," Management Science, INFORMS, vol. 9(1), pages 16-24, October.
    7. Yinyu Ye, 2011. "The Simplex and Policy-Iteration Methods Are Strongly Polynomial for the Markov Decision Problem with a Fixed Discount Rate," Mathematics of Operations Research, INFORMS, vol. 36(4), pages 593-603, November.
    8. Alexander Zadorojniy & Guy Even & Adam Shwartz, 2009. "A Strongly Polynomial Algorithm for Controlled Queues," Mathematics of Operations Research, INFORMS, vol. 34(4), pages 992-1007, November.
    9. Yinyu Ye, 2005. "A New Complexity Result on Solving the Markov Decision Problem," Mathematics of Operations Research, INFORMS, vol. 30(3), pages 733-749, August.
    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. Alexander Zadorojniy & Guy Even & Adam Shwartz, 2009. "A Strongly Polynomial Algorithm for Controlled Queues," Mathematics of Operations Research, INFORMS, vol. 34(4), pages 992-1007, November.
    2. Ian Post & Yinyu Ye, 2015. "The Simplex Method is Strongly Polynomial for Deterministic Markov Decision Processes," Mathematics of Operations Research, INFORMS, vol. 40(4), pages 859-868, October.
    3. Archis Ghate & Robert L. Smith, 2013. "A Linear Programming Approach to Nonstationary Infinite-Horizon Markov Decision Processes," Operations Research, INFORMS, vol. 61(2), pages 413-425, April.
    4. Guillot, Matthieu & Stauffer, Gautier, 2020. "The Stochastic Shortest Path Problem: A polyhedral combinatorics perspective," European Journal of Operational Research, Elsevier, vol. 285(1), pages 148-158.
    5. Lodewijk Kallenberg, 2013. "Derman’s book as inspiration: some results on LP for MDPs," Annals of Operations Research, Springer, vol. 208(1), pages 63-94, September.
    6. Mengdi Wang, 2020. "Randomized Linear Programming Solves the Markov Decision Problem in Nearly Linear (Sometimes Sublinear) Time," Mathematics of Operations Research, INFORMS, vol. 45(2), pages 517-546, May.
    7. B. Curtis Eaves & Arthur F. Veinott, 2014. "Maximum-Stopping-Value Policies in Finite Markov Population Decision Chains," Mathematics of Operations Research, INFORMS, vol. 39(3), pages 597-606, August.
    8. K. Helmes & R. H. Stockbridge, 2000. "Numerical Comparison of Controls and Verification of Optimality for Stochastic Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 106(1), pages 107-127, July.
    9. Michael O’Sullivan & Arthur F. Veinott, Jr., 2017. "Polynomial-Time Computation of Strong and n -Present-Value Optimal Policies in Markov Decision Chains," Mathematics of Operations Research, INFORMS, vol. 42(3), pages 577-598, August.
    10. Yinyu Ye, 2011. "The Simplex and Policy-Iteration Methods Are Strongly Polynomial for the Markov Decision Problem with a Fixed Discount Rate," Mathematics of Operations Research, INFORMS, vol. 36(4), pages 593-603, November.
    11. Eugene A. Feinberg & Jefferson Huang, 2019. "On the reduction of total‐cost and average‐cost MDPs to discounted MDPs," Naval Research Logistics (NRL), John Wiley & Sons, vol. 66(1), pages 38-56, February.
    12. José Niño-Mora, 2006. "Restless Bandit Marginal Productivity Indices, Diminishing Returns, and Optimal Control of Make-to-Order/Make-to-Stock M/G/1 Queues," Mathematics of Operations Research, INFORMS, vol. 31(1), pages 50-84, February.
    13. Oguzhan Alagoz & Lisa M. Maillart & Andrew J. Schaefer & Mark S. Roberts, 2007. "Determining the Acceptance of Cadaveric Livers Using an Implicit Model of the Waiting List," Operations Research, INFORMS, vol. 55(1), pages 24-36, February.
    14. Ting Pong & Hao Sun & Ningchuan Wang & Henry Wolkowicz, 2016. "Eigenvalue, quadratic programming, and semidefinite programming relaxations for a cut minimization problem," Computational Optimization and Applications, Springer, vol. 63(2), pages 333-364, March.
    15. Allahviranloo, Mahdieh & Recker, Will, 2013. "Daily activity pattern recognition by using support vector machines with multiple classes," Transportation Research Part B: Methodological, Elsevier, vol. 58(C), pages 16-43.
    16. R. B. Bapat & S. K. Neogy, 2016. "On a quadratic programming problem involving distances in trees," Annals of Operations Research, Springer, vol. 243(1), pages 365-373, August.
    17. Höfferl, F. & Steinschorn, D., 2009. "A dynamic programming extension to the steady state refinery-LP," European Journal of Operational Research, Elsevier, vol. 197(2), pages 465-474, September.
    18. Amitai Armon & Iftah Gamzu & Danny Segev, 2014. "Mobile facility location: combinatorial filtering via weighted occupancy," Journal of Combinatorial Optimization, Springer, vol. 28(2), pages 358-375, August.
    19. Balaji Gopalakrishnan & Seunghyun Kong & Earl Barnes & Ellis Johnson & Joel Sokol, 2011. "A least-squares minimum-cost network flow algorithm," Annals of Operations Research, Springer, vol. 186(1), pages 119-140, June.
    20. Xiuli Chao & Frank Y. Chen, 2005. "An Optimal Production and Shutdown Strategy when a Supplier Offers an Incentive Program," Manufacturing & Service Operations Management, INFORMS, vol. 7(2), pages 130-143, March.

    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:spr:annopr:v:201:y:2012:i:1:p:159-167:10.1007/s10479-012-1199-x. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.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.