IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v62y2015i4p643-659.html
   My bibliography  Save this article

Duality and robust duality for special nonconvex homogeneous quadratic programming under certainty and uncertainty environment

Author

Listed:
  • Yanjun Wang
  • Ruizhi Shi
  • Jianming Shi

Abstract

In this paper, we discuss a kind of special nonconvex homogenous quadratic programming (HQP) and the methods to solve the HQP in an environment with certainty or uncertainty. In an environment with certainty, we first establish a strong duality between the HQP and its Lagrange dual problem, with the help of the fact that the Lagrange dual problem is equivalent to a convex semidefinite programming (SDP). Then we obtain a global solution to the HQP by solving the convex SDP. Furthermore, in an environment with uncertainty, we formulate the robust counterpart of the HQP to cope with uncertainty. We also establish the robust strong duality between the robust counterpart and its optimistic counterpart under a mild assumption. Since the counterpart is equivalent to a convex SDP under the same assumption, we can obtain a global solution to the robust counterpart by solving the convex SDP under the same assumption. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Yanjun Wang & Ruizhi Shi & Jianming Shi, 2015. "Duality and robust duality for special nonconvex homogeneous quadratic programming under certainty and uncertainty environment," Journal of Global Optimization, Springer, vol. 62(4), pages 643-659, August.
    • Handle: RePEc:spr:jglopt:v:62:y:2015:i:4:p:643-659
    DOI: 10.1007/s10898-015-0281-8
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10898-015-0281-8
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10898-015-0281-8?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. V. Jeyakumar & G. Y. Li, 2011. "Robust Duality for Fractional Programming Problems with Constraint-Wise Data Uncertainty," Journal of Optimization Theory and Applications, Springer, vol. 151(2), pages 292-303, November.
    2. Alexander Shapiro, 1982. "Rank-reducibility of a symmetric matrix and sampling theory of minimum trace factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 47(2), pages 187-199, June.
    3. Gábor Pataki, 1998. "On the Rank of Extreme Matrices in Semidefinite Programs and the Multiplicity of Optimal Eigenvalues," Mathematics of Operations Research, INFORMS, vol. 23(2), pages 339-358, May.
    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. Nguyen Dinh & Miguel Angel Goberna & Marco Antonio López & Michel Volle, 2017. "A Unifying Approach to Robust Convex Infinite Optimization Duality," Journal of Optimization Theory and Applications, Springer, vol. 174(3), pages 650-685, September.

    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. Jos Berge & Henk Kiers, 1991. "A numerical approach to the approximate and the exact minimum rank of a covariance matrix," Psychometrika, Springer;The Psychometric Society, vol. 56(2), pages 309-315, June.
    2. Jeyakumar, V. & Li, G.Y. & Srisatkunarajah, S., 2013. "Strong duality for robust minimax fractional programming problems," European Journal of Operational Research, Elsevier, vol. 228(2), pages 331-336.
    3. Xinzhen Zhang & Chen Ling & Liqun Qi, 2011. "Semidefinite relaxation bounds for bi-quadratic optimization problems with quadratic constraints," Journal of Global Optimization, Springer, vol. 49(2), pages 293-311, February.
    4. Bomze, Immanuel M. & Gabl, Markus, 2023. "Optimization under uncertainty and risk: Quadratic and copositive approaches," European Journal of Operational Research, Elsevier, vol. 310(2), pages 449-476.
    5. Stefan Sremac & Fei Wang & Henry Wolkowicz & Lucas Pettersson, 2019. "Noisy Euclidean distance matrix completion with a single missing node," Journal of Global Optimization, Springer, vol. 75(4), pages 973-1002, December.
    6. Madeleine Udell & Stephen Boyd, 2016. "Bounding duality gap for separable problems with linear constraints," Computational Optimization and Applications, Springer, vol. 64(2), pages 355-378, June.
    7. Shapiro, Alexander, 2009. "Asymptotic normality of test statistics under alternative hypotheses," Journal of Multivariate Analysis, Elsevier, vol. 100(5), pages 936-945, May.
    8. Kaufmann, Sylvia & Schumacher, Christian, 2019. "Bayesian estimation of sparse dynamic factor models with order-independent and ex-post mode identification," Journal of Econometrics, Elsevier, vol. 210(1), pages 116-134.
    9. Anthony Man-Cho So & Yinyu Ye & Jiawei Zhang, 2008. "A Unified Theorem on SDP Rank Reduction," Mathematics of Operations Research, INFORMS, vol. 33(4), pages 910-920, November.
    10. Knott, Martin, 2005. "A measure of independence for a multivariate normal distribution and some connections with factor analysis," Journal of Multivariate Analysis, Elsevier, vol. 96(2), pages 374-383, October.
    11. Immanuel Bomze & Markus Gabl, 2021. "Interplay of non-convex quadratically constrained problems with adjustable robust optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 93(1), pages 115-151, February.
    12. Jiawei Chen & Suliman Al-Homidan & Qamrul Hasan Ansari & Jun Li & Yibing Lv, 2021. "Robust Necessary Optimality Conditions for Nondifferentiable Complex Fractional Programming with Uncertain Data," Journal of Optimization Theory and Applications, Springer, vol. 189(1), pages 221-243, April.
    13. Bai, Jushan & Ng, Serena, 2019. "Rank regularized estimation of approximate factor models," Journal of Econometrics, Elsevier, vol. 212(1), pages 78-96.
    14. Jos F. Sturm & Shuzhong Zhang, 2003. "On Cones of Nonnegative Quadratic Functions," Mathematics of Operations Research, INFORMS, vol. 28(2), pages 246-267, May.
    15. Im, Haesol & Wolkowicz, Henry, 2023. "Revisiting degeneracy, strict feasibility, stability, in linear programming," European Journal of Operational Research, Elsevier, vol. 310(2), pages 495-510.
    16. Debdas Ghosh, 2016. "A Newton method for capturing efficient solutions of interval optimization problems," OPSEARCH, Springer;Operational Research Society of India, vol. 53(3), pages 648-665, September.
    17. Boris Defourny & Ilya O. Ryzhov & Warren B. Powell, 2015. "Optimal Information Blending with Measurements in the L 2 Sphere," Mathematics of Operations Research, INFORMS, vol. 40(4), pages 1060-1088, October.
    18. Alexander Shapiro & Jos Berge, 2000. "The asymptotic bias of minimum trace factor analysis, with applications to the greatest lower bound to reliability," Psychometrika, Springer;The Psychometric Society, vol. 65(3), pages 413-425, September.
    19. Wenbao Ai & Yongwei Huang & Shuzhong Zhang, 2008. "On the Low Rank Solutions for Linear Matrix Inequalities," Mathematics of Operations Research, INFORMS, vol. 33(4), pages 965-975, November.
    20. Tyler Hunt & Peter Bentler, 2015. "Quantile Lower Bounds to Reliability Based on Locally Optimal Splits," Psychometrika, Springer;The Psychometric Society, vol. 80(1), pages 182-195, 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:jglopt:v:62:y:2015:i:4:p:643-659. 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.