IDEAS home Printed from https://ideas.repec.org/a/spr/mathme/v72y2010i2p187-204.html
   My bibliography  Save this article

Unboundedness in reverse convex and concave integer programming

Author

Listed:
  • Wiesława Obuchowska

Abstract

In this paper we are concerned with the problem of unboundedness and existence of an optimal solution in reverse convex and concave integer optimization problems. In particular, we present necessary and sufficient conditions for existence of an upper bound for a convex objective function defined over the feasible region contained in $${\mathbb{Z}^n}$$ . The conditions for boundedness are provided in a form of an implementable algorithm, showing that for the considered class of functions, the integer programming problem is unbounded if and only if the associated continuous problem is unbounded. We also address the problem of boundedness in the global optimization problem of maximizing a convex function over a set of integers contained in a convex and unbounded region. It is shown in the paper that in both types of integer programming problems, the objective function is either unbounded from above, or it attains its maximum at a feasible integer point. Copyright Springer-Verlag 2010

Suggested Citation

  • Wiesława Obuchowska, 2010. "Unboundedness in reverse convex and concave integer programming," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 72(2), pages 187-204, October.
    • Handle: RePEc:spr:mathme:v:72:y:2010:i:2:p:187-204
    DOI: 10.1007/s00186-010-0315-4
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s00186-010-0315-4
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s00186-010-0315-4?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. Duan Li & Xiaoling Sun, 2006. "Nonlinear Integer Programming," International Series in Operations Research and Management Science, Springer, number 978-0-387-32995-6, December.
    2. Wiesława Obuchowska, 2007. "Conditions for boundedness in concave programming under reverse convex and convex constraints," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 65(2), pages 261-279, April.
    3. Wiesława Obuchowska, 2010. "Minimal infeasible constraint sets in convex integer programs," Journal of Global Optimization, Springer, vol. 46(3), pages 423-433, March.
    4. Obuchowska, W. T. & Murty, K. G., 2001. "Cone of recession and unboundedness of convex functions," European Journal of Operational Research, Elsevier, vol. 133(2), pages 409-415, January.
    5. Caron, R. J. & Obuchowska, W., 1992. "Unboundedness of a convex quadratic function subject to concave and convex quadratic constraints," European Journal of Operational Research, Elsevier, vol. 63(1), pages 114-123, November.
    6. Wiesława Obuchowska, 2008. "On boundedness of (quasi-)convex integer optimization problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 68(3), pages 445-467, December.
    7. Caron, Richard J. & Obuchowska, Wieslawa T., 1995. "An algorithm to determine boundedness of quadratically constrained convex quadratic programmes," European Journal of Operational Research, Elsevier, vol. 80(2), pages 431-438, January.
    8. H. Tuy & T. V. Thieu & Ng. Q. Thai, 1985. "A Conical Algorithm for Globally Minimizing a Concave Function Over a Closed Convex Set," Mathematics of Operations Research, INFORMS, vol. 10(3), pages 498-514, August.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Wiesława T. Obuchowska, 2015. "Irreducible Infeasible Sets in Convex Mixed-Integer Programs," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 747-766, September.
    2. Obuchowska, Wiesława T., 2014. "Feasible partition problem in reverse convex and convex mixed-integer programming," European Journal of Operational Research, Elsevier, vol. 235(1), pages 129-137.
    3. Obuchowska, Wiesława T., 2012. "Feasibility in reverse convex mixed-integer programming," European Journal of Operational Research, Elsevier, vol. 218(1), pages 58-67.

    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. Obuchowska, Wiesława T., 2014. "Feasible partition problem in reverse convex and convex mixed-integer programming," European Journal of Operational Research, Elsevier, vol. 235(1), pages 129-137.
    2. Obuchowska, Wiesława T., 2012. "Feasibility in reverse convex mixed-integer programming," European Journal of Operational Research, Elsevier, vol. 218(1), pages 58-67.
    3. Wiesława Obuchowska, 2007. "Conditions for boundedness in concave programming under reverse convex and convex constraints," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 65(2), pages 261-279, April.
    4. Wiesława T. Obuchowska, 2015. "Irreducible Infeasible Sets in Convex Mixed-Integer Programs," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 747-766, September.
    5. Caron, Richard J. & Obuchowska, Wieslawa T., 1996. "Quadratically constrained convex quadratic programmes: faulty feasible regions," European Journal of Operational Research, Elsevier, vol. 94(1), pages 134-142, October.
    6. Walter Briec & Laurent Cavaignac & Kristiaan Kerstens, 2020. "Input Efficiency Measures: A Generalized, Encompassing Formulation," Operations Research, INFORMS, vol. 68(6), pages 1836-1849, November.
    7. Guo, Jian-Xin & Huang, Chen, 2020. "Feasible roadmap for CCS retrofit of coal-based power plants to reduce Chinese carbon emissions by 2050," Applied Energy, Elsevier, vol. 259(C).
    8. Wiesława Obuchowska, 2008. "On boundedness of (quasi-)convex integer optimization problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 68(3), pages 445-467, December.
    9. Cascón, J.M. & González-Arteaga, T. & de Andrés Calle, R., 2019. "Reaching social consensus family budgets: The Spanish case," Omega, Elsevier, vol. 86(C), pages 28-41.
    10. Fatima Bellahcene, 2019. "Application of the polyblock method to special integer chance constrained problem," Operations Research and Decisions, Wroclaw University of Science and Technology, Faculty of Management, vol. 29(4), pages 23-40.
    11. Chunli Liu & Jianjun Gao, 2015. "A polynomial case of convex integer quadratic programming problems with box integer constraints," Journal of Global Optimization, Springer, vol. 62(4), pages 661-674, August.
    12. Obuchowska, W. T. & Murty, K. G., 2001. "Cone of recession and unboundedness of convex functions," European Journal of Operational Research, Elsevier, vol. 133(2), pages 409-415, January.
    13. L. Escudero & M. Garín & G. Pérez & A. Unzueta, 2012. "Lagrangian Decomposition for large-scale two-stage stochastic mixed 0-1 problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(2), pages 347-374, July.
    14. Kouhei Harada, 2021. "A Feasibility-Ensured Lagrangian Heuristic for General Decomposable Problems," SN Operations Research Forum, Springer, vol. 2(4), pages 1-26, December.
    15. Bin Zhang & Bo Chen, 2012. "Heuristic And Exact Solution Method For Convex Nonlinear Knapsack Problem," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 29(05), pages 1-14.
    16. Obuchowska, Wieslawa T., 1998. "Infeasibility analysis for systems of quadratic convex inequalities," European Journal of Operational Research, Elsevier, vol. 107(3), pages 633-643, June.
    17. Lin, Yun Hui & Wang, Yuan & Lee, Loo Hay & Chew, Ek Peng, 2022. "Omnichannel facility location and fulfillment optimization," Transportation Research Part B: Methodological, Elsevier, vol. 163(C), pages 187-209.
    18. Federico Della Croce & Dominique Quadri, 2012. "Improving an exact approach for solving separable integer quadratic knapsack problems," Journal of Combinatorial Optimization, Springer, vol. 23(1), pages 21-28, January.
    19. Alidaee, Bahram, 2014. "Zero duality gap in surrogate constraint optimization: A concise review of models," European Journal of Operational Research, Elsevier, vol. 232(2), pages 241-248.

    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:mathme:v:72:y:2010:i:2:p:187-204. 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.