IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v95y1997i2d10.1023_a1022687222060.html
   My bibliography  Save this article

Lagrange Duality and Partitioning Techniques in Nonconvex Global Optimization

Author

Listed:
  • M. Dür

    (University of Trier)

  • R. Horst

    (University of Trier)

Abstract

It is shown that, for very general classes of nonconvex global optimization problems, the duality gap obtained by solving a corresponding Lagrangian dual in reduced to zero in the limit when combined with suitably refined partitioning of the feasible set. A similar result holds for partly convex problems where exhaustive partitioning is applied only in the space of nonconvex variables. Applications include branch-and-bound approaches for linearly constrained problems where convex envelopes can be computed, certain generalized bilinear problems, linearly constrained optimization of the sum of ratios of affine functions, and concave minimization under reverse convex constraints.

Suggested Citation

  • M. Dür & R. Horst, 1997. "Lagrange Duality and Partitioning Techniques in Nonconvex Global Optimization," Journal of Optimization Theory and Applications, Springer, vol. 95(2), pages 347-369, November.
    • Handle: RePEc:spr:joptap:v:95:y:1997:i:2:d:10.1023_a:1022687222060
    DOI: 10.1023/A:1022687222060
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1023/A:1022687222060
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1023/A:1022687222060?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. M. Pappalardo, 1986. "On the Duality Gap in Nonconvex Optimization," Mathematics of Operations Research, INFORMS, vol. 11(1), pages 30-35, February.
    2. Faiz A. Al-Khayyal & James E. Falk, 1983. "Jointly Constrained Biconvex Programming," Mathematics of Operations Research, INFORMS, vol. 8(2), pages 273-286, May.
    3. J. P. Aubin & I. Ekeland, 1976. "Estimates of the Duality Gap in Nonconvex Optimization," Mathematics of Operations Research, INFORMS, vol. 1(3), pages 225-245, 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. Rohit Kannan & Paul I. Barton, 2018. "Convergence-order analysis of branch-and-bound algorithms for constrained problems," Journal of Global Optimization, Springer, vol. 71(4), pages 753-813, August.
    2. Moslem Zamani, 2019. "A new algorithm for concave quadratic programming," Journal of Global Optimization, Springer, vol. 75(3), pages 655-681, November.
    3. Charles Audet & Jack Brimberg & Pierre Hansen & Sébastien Le Digabel & Nenad Mladenovi'{c}, 2004. "Pooling Problem: Alternate Formulations and Solution Methods," Management Science, INFORMS, vol. 50(6), pages 761-776, June.
    4. N.V. Thoai, 2002. "Convergence and Application of a Decomposition Method Using Duality Bounds for Nonconvex Global Optimization," Journal of Optimization Theory and Applications, Springer, vol. 113(1), pages 165-193, April.
    5. Marco Locatelli & Fabio Schoen, 2012. "On the relation between concavity cuts and the surrogate dual for convex maximization problems," Journal of Global Optimization, Springer, vol. 52(3), pages 411-421, March.
    6. Yankai Cao & Victor M. Zavala, 2019. "A scalable global optimization algorithm for stochastic nonlinear programs," Journal of Global Optimization, Springer, vol. 75(2), pages 393-416, October.
    7. N. V. Thoai, 2000. "Duality Bound Method for the General Quadratic Programming Problem with Quadratic Constraints," Journal of Optimization Theory and Applications, Springer, vol. 107(2), pages 331-354, November.
    8. Benson, Harold P., 2007. "A simplicial branch and bound duality-bounds algorithm for the linear sum-of-ratios problem," European Journal of Operational Research, Elsevier, vol. 182(2), pages 597-611, October.

    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. Borglin, Anders & Flåm, Sjur Didrik, 2007. "Risk exchange as a market or production game," Working Papers in Economics 09/07, University of Bergen, Department of Economics.
    2. N. V. Thoai, 2000. "Duality Bound Method for the General Quadratic Programming Problem with Quadratic Constraints," Journal of Optimization Theory and Applications, Springer, vol. 107(2), pages 331-354, November.
    3. Evrim Dalkiran & Hanif Sherali, 2013. "Theoretical filtering of RLT bound-factor constraints for solving polynomial programming problems to global optimality," Journal of Global Optimization, Springer, vol. 57(4), pages 1147-1172, December.
    4. Gabriele Eichfelder & Peter Kirst & Laura Meng & Oliver Stein, 2021. "A general branch-and-bound framework for continuous global multiobjective optimization," Journal of Global Optimization, Springer, vol. 80(1), pages 195-227, May.
    5. S. Flåm & L. Koutsougeras, 2010. "Private information, transferable utility, and the core," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 42(3), pages 591-609, March.
    6. Kouhei Harada, 2021. "A Feasibility-Ensured Lagrangian Heuristic for General Decomposable Problems," SN Operations Research Forum, Springer, vol. 2(4), pages 1-26, December.
    7. Sonia Cafieri & Jon Lee & Leo Liberti, 2010. "On convex relaxations of quadrilinear terms," Journal of Global Optimization, Springer, vol. 47(4), pages 661-685, August.
    8. Keith Zorn & Nikolaos Sahinidis, 2014. "Global optimization of general nonconvex problems with intermediate polynomial substructures," Journal of Global Optimization, Springer, vol. 59(2), pages 673-693, July.
    9. Tsao, Yu-Chung & Lu, Jye-Chyi & An, Na & Al-Khayyal, Faiz & Lu, Richard W. & Han, Guanghua, 2014. "Retailer shelf-space management with trade allowance: A Stackelberg game between retailer and manufacturers," International Journal of Production Economics, Elsevier, vol. 148(C), pages 133-144.
    10. Al-Khayyal, Faiz & Hwang, Seung-June, 2007. "Inventory constrained maritime routing and scheduling for multi-commodity liquid bulk, Part I: Applications and model," European Journal of Operational Research, Elsevier, vol. 176(1), pages 106-130, January.
    11. Daniel Adelman & Adam J. Mersereau, 2008. "Relaxations of Weakly Coupled Stochastic Dynamic Programs," Operations Research, INFORMS, vol. 56(3), pages 712-727, June.
    12. Christis Katsouris, 2023. "Optimal Estimation Methodologies for Panel Data Regression Models," Papers 2311.03471, arXiv.org, revised Nov 2023.
    13. Manuel Ruiz & Olivier Briant & Jean-Maurice Clochard & Bernard Penz, 2013. "Large-scale standard pooling problems with constrained pools and fixed demands," Journal of Global Optimization, Springer, vol. 56(3), pages 939-956, July.
    14. Marco Locatelli, 2016. "Polyhedral subdivisions and functional forms for the convex envelopes of bilinear, fractional and other bivariate functions over general polytopes," Journal of Global Optimization, Springer, vol. 66(4), pages 629-668, December.
    15. Hezhi Luo & Xiaodong Ding & Jiming Peng & Rujun Jiang & Duan Li, 2021. "Complexity Results and Effective Algorithms for Worst-Case Linear Optimization Under Uncertainties," INFORMS Journal on Computing, INFORMS, vol. 33(1), pages 180-197, January.
    16. Kristianto, Yohanes & Helo, Petri, 2014. "Product architecture modularity implications for operations economy of green supply chains," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 70(C), pages 128-145.
    17. Juan P. Ruiz & Ignacio E. Grossmann, 2017. "Global optimization of non-convex generalized disjunctive programs: a review on reformulations and relaxation techniques," Journal of Global Optimization, Springer, vol. 67(1), pages 43-58, January.
    18. Wang, Yadong & Meng, Qiang & Du, Yuquan, 2015. "Liner container seasonal shipping revenue management," Transportation Research Part B: Methodological, Elsevier, vol. 82(C), pages 141-161.
    19. Luathep, Paramet & Sumalee, Agachai & Lam, William H.K. & Li, Zhi-Chun & Lo, Hong K., 2011. "Global optimization method for mixed transportation network design problem: A mixed-integer linear programming approach," Transportation Research Part B: Methodological, Elsevier, vol. 45(5), pages 808-827, June.
    20. E Bustamante-Cedeño & S Arora, 2008. "Stochastic and minimum regret formulations for transmission network expansion planning under uncertainties," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 59(11), pages 1547-1556, November.

    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:joptap:v:95:y:1997:i:2:d:10.1023_a:1022687222060. 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.