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. 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.
    2. 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.
    3. Moslem Zamani, 2019. "A new algorithm for concave quadratic programming," Journal of Global Optimization, Springer, vol. 75(3), pages 655-681, November.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.

    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, 2007. "Risk Exchange as a Market or Production Game," Working Papers 2007:16, Lund University, Department of Economics.
    2. İhsan Yanıkoğlu & Erinç Albey & Serkan Okçuoğlu, 2022. "Robust Parameter Design and Optimization for Quality Engineering," SN Operations Research Forum, Springer, vol. 3(1), pages 1-36, March.
    3. Eli Towle & James Luedtke, 2018. "New solution approaches for the maximum-reliability stochastic network interdiction problem," Computational Management Science, Springer, vol. 15(3), pages 455-477, October.
    4. 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.
    5. Radu Baltean-Lugojan & Ruth Misener, 2018. "Piecewise parametric structure in the pooling problem: from sparse strongly-polynomial solutions to NP-hardness," Journal of Global Optimization, Springer, vol. 71(4), pages 655-690, August.
    6. 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.
    7. M. M. Faruque Hasan, 2018. "An edge-concave underestimator for the global optimization of twice-differentiable nonconvex problems," Journal of Global Optimization, Springer, vol. 71(4), pages 735-752, August.
    8. Wu, Haoyu & Tang, Ao, 2024. "An even tighter bound for the Shapley–Folkman–Starr theorem," Journal of Mathematical Economics, Elsevier, vol. 114(C).
    9. 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.
    10. 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.
    11. Kouhei Harada, 2021. "A Feasibility-Ensured Lagrangian Heuristic for General Decomposable Problems," SN Operations Research Forum, Springer, vol. 2(4), pages 1-26, December.
    12. 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.
    13. 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.
    14. 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.
    15. 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.
    16. Daniel Adelman & Adam J. Mersereau, 2008. "Relaxations of Weakly Coupled Stochastic Dynamic Programs," Operations Research, INFORMS, vol. 56(3), pages 712-727, June.
    17. Christis Katsouris, 2023. "Optimal Estimation Methodologies for Panel Data Regression Models," Papers 2311.03471, arXiv.org, revised Nov 2023.
    18. Marcia Fampa & Jon Lee, 2021. "Convexification of bilinear forms through non-symmetric lifting," Journal of Global Optimization, Springer, vol. 80(2), pages 287-305, June.
    19. 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.
    20. Michelle L. Blom & Christina N. Burt & Adrian R. Pearce & Peter J. Stuckey, 2014. "A Decomposition-Based Heuristic for Collaborative Scheduling in a Network of Open-Pit Mines," INFORMS Journal on Computing, INFORMS, vol. 26(4), pages 658-676, 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.