IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v55y2013i1p21-47.html
   My bibliography  Save this article

SDP reformulation for robust optimization problems based on nonconvex QP duality

Author

Listed:
  • Ryoichi Nishimura
  • Shunsuke Hayashi
  • Masao Fukushima

Abstract

In a real situation, optimization problems often involve uncertain parameters. Robust optimization is one of distribution-free methodologies based on worst-case analyses for handling such problems. In this paper, we first focus on a special class of uncertain linear programs (LPs). Applying the duality theory for nonconvex quadratic programs (QPs), we reformulate the robust counterpart as a semidefinite program (SDP) and show the equivalence property under mild assumptions. We also apply the same technique to the uncertain second-order cone programs (SOCPs) with “single” (not side-wise) ellipsoidal uncertainty. Then we derive similar results on the reformulation and the equivalence property. In the numerical experiments, we solve some test problems to demonstrate the efficiency of our reformulation approach. Especially, we compare our approach with another recent method based on Hildebrand’s Lorentz positivity. Copyright Springer Science+Business Media New York 2013

Suggested Citation

  • Ryoichi Nishimura & Shunsuke Hayashi & Masao Fukushima, 2013. "SDP reformulation for robust optimization problems based on nonconvex QP duality," Computational Optimization and Applications, Springer, vol. 55(1), pages 21-47, May.
    • Handle: RePEc:spr:coopap:v:55:y:2013:i:1:p:21-47
    DOI: 10.1007/s10589-012-9520-9
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10589-012-9520-9
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10589-012-9520-9?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. Dimitris Bertsimas & Melvyn Sim, 2004. "The Price of Robustness," Operations Research, INFORMS, vol. 52(1), pages 35-53, February.
    2. Lopez, Marco & Still, Georg, 2007. "Semi-infinite programming," European Journal of Operational Research, Elsevier, vol. 180(2), pages 491-518, July.
    3. Zymler, Steve & Rustem, Berç & Kuhn, Daniel, 2011. "Robust portfolio optimization with derivative insurance guarantees," European Journal of Operational Research, Elsevier, vol. 210(2), pages 410-424, April.
    4. Dimitris Bertsimas & Aurélie Thiele, 2006. "A Robust Optimization Approach to Inventory Theory," Operations Research, INFORMS, vol. 54(1), pages 150-168, February.
    5. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(1), pages 193-194, February.
    6. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(2), pages 541-545, April.
    7. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(4), pages 1007-1017, August.
    8. Laurent El Ghaoui & Maksim Oks & Francois Oustry, 2003. "Worst-Case Value-At-Risk and Robust Portfolio Optimization: A Conic Programming Approach," Operations Research, INFORMS, vol. 51(4), pages 543-556, August.
    9. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(6), pages 1461-1465, December.
    10. Shushang Zhu & Masao Fukushima, 2009. "Worst-Case Conditional Value-at-Risk with Application to Robust Portfolio Management," Operations Research, INFORMS, vol. 57(5), pages 1155-1168, October.
    11. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(5), pages 1273-1289, October.
    12. Huang, Dashan & Zhu, Shu-Shang & Fabozzi, Frank J. & Fukushima, Masao, 2008. "Portfolio selection with uncertain exit time: A robust CVaR approach," Journal of Economic Dynamics and Control, Elsevier, vol. 32(2), pages 594-623, February.
    13. ,, 2002. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 18(3), pages 819-821, June.
    14. A. Ben-Tal & A. Nemirovski, 1998. "Robust Convex Optimization," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 769-805, November.
    15. D. Goldfarb & G. Iyengar, 2003. "Robust Portfolio Selection Problems," Mathematics of Operations Research, INFORMS, vol. 28(1), pages 1-38, February.
    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. Crespi, Giovanni P. & Kuroiwa, Daishi & Rocca, Matteo, 2018. "Robust optimization: Sensitivity to uncertainty in scalar and vector cases, with applications," Operations Research Perspectives, Elsevier, vol. 5(C), pages 113-119.
    2. Giovanni P. Crespi & Daishi Kuroiwa & Matteo Rocca, 2017. "Quasiconvexity of set-valued maps assures well-posedness of robust vector optimization," Annals of Operations Research, Springer, vol. 251(1), pages 89-104, April.

    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. Soleimanian, Azam & Salmani Jajaei, Ghasemali, 2013. "Robust nonlinear optimization with conic representable uncertainty set," European Journal of Operational Research, Elsevier, vol. 228(2), pages 337-344.
    2. Nikulin, Yury, 2006. "Robustness in combinatorial optimization and scheduling theory: An extended annotated bibliography," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 606, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.
    3. Garud N. Iyengar, 2005. "Robust Dynamic Programming," Mathematics of Operations Research, INFORMS, vol. 30(2), pages 257-280, May.
    4. Dan A. Iancu & Mayank Sharma & Maxim Sviridenko, 2013. "Supermodularity and Affine Policies in Dynamic Robust Optimization," Operations Research, INFORMS, vol. 61(4), pages 941-956, August.
    5. Luo, Fengqiao & Mehrotra, Sanjay, 2019. "Decomposition algorithm for distributionally robust optimization using Wasserstein metric with an application to a class of regression models," European Journal of Operational Research, Elsevier, vol. 278(1), pages 20-35.
    6. Areesh Mittal & Can Gokalp & Grani A. Hanasusanto, 2020. "Robust Quadratic Programming with Mixed-Integer Uncertainty," INFORMS Journal on Computing, INFORMS, vol. 32(2), pages 201-218, April.
    7. Odellia Boni & Aharon Ben-Tal, 2008. "Adjustable robust counterpart of conic quadratic problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 68(2), pages 211-233, October.
    8. Kürşad Derinkuyu & Mustafa Pınar, 2006. "On the S-procedure and Some Variants," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 64(1), pages 55-77, August.
    9. Artur Alves Pessoa & Michael Poss, 2015. "Robust Network Design with Uncertain Outsourcing Cost," INFORMS Journal on Computing, INFORMS, vol. 27(3), pages 507-524, August.
    10. Dimitris Bertsimas & Dan A. Iancu & Pablo A. Parrilo, 2010. "Optimality of Affine Policies in Multistage Robust Optimization," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 363-394, May.
    11. Alireza Ghahtarani & Ahmed Saif & Alireza Ghasemi, 2022. "Robust portfolio selection problems: a comprehensive review," Operational Research, Springer, vol. 22(4), pages 3203-3264, September.
    12. Ralf Werner, 2008. "Cascading: an adjusted exchange method for robust conic programming," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 16(2), pages 179-189, June.
    13. Panos Xidonas & Ralph Steuer & Christis Hassapis, 2020. "Robust portfolio optimization: a categorized bibliographic review," Annals of Operations Research, Springer, vol. 292(1), pages 533-552, September.
    14. Alireza Ghahtarani & Ahmed Saif & Alireza Ghasemi, 2021. "Robust Portfolio Selection Problems: A Comprehensive Review," Papers 2103.13806, arXiv.org, revised Jan 2022.
    15. Gülpınar, Nalân & Çanakoḡlu, Ethem, 2017. "Robust portfolio selection problem under temperature uncertainty," European Journal of Operational Research, Elsevier, vol. 256(2), pages 500-523.
    16. Claudia García-García & Catalina B. García-García & Román Salmerón, 2021. "Confronting collinearity in environmental regression models: evidence from world data," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 30(3), pages 895-926, September.
    17. Cambier, Adrien & Chardy, Matthieu & Figueiredo, Rosa & Ouorou, Adam & Poss, Michael, 2022. "Optimizing subscriber migrations for a telecommunication operator in uncertain context," European Journal of Operational Research, Elsevier, vol. 298(1), pages 308-321.
    18. Libura, Marek, 2007. "On the adjustment problem for linear programs," European Journal of Operational Research, Elsevier, vol. 183(1), pages 125-134, November.
    19. Christophe Loussouarn & Carine Franc & Yann Videau & Julien Mousquès, 2021. "Can General Practitioners Be More Productive? The Impact of Teamwork and Cooperation with Nurses on GP Activities," Health Economics, John Wiley & Sons, Ltd., vol. 30(3), pages 680-698, March.
    20. Tschakert, Petra, 2016. "Shifting Discourses of Vilification and the Taming of Unruly Mining Landscapes in Ghana," World Development, Elsevier, vol. 86(C), pages 123-132.

    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:coopap:v:55:y:2013:i:1:p:21-47. 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.