IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v197y2023i2d10.1007_s10957-023-02184-6.html
   My bibliography  Save this article

Characterizing a Class of Robust Vector Polynomial Optimization via Sum of Squares Conditions

Author

Listed:
  • Xiangkai Sun

    (Chongqing Technology and Business University)

  • Wen Tan

    (Chongqing Technology and Business University)

  • Kok Lay Teo

    (Sunway University)

Abstract

This paper deals with an SOS-convex (sum of squares convex) polynomial optimization problem with spectrahedral uncertain data in both the objective and constraints. By using a robust-type characteristic cone constraint qualification, we first obtain necessary and sufficient conditions for robust weakly efficient solutions of this uncertain SOS-convex polynomial optimization problem in terms of sum of squares conditions and linear matrix inequalities. Then, we propose a relaxation dual problem for this uncertain SOS-convex polynomial optimization problem and explore weak and strong duality properties between them. Moreover, we give a numerical example to show that the relaxation dual problem can be reformulated as a semidefinite linear programming problem.

Suggested Citation

  • Xiangkai Sun & Wen Tan & Kok Lay Teo, 2023. "Characterizing a Class of Robust Vector Polynomial Optimization via Sum of Squares Conditions," Journal of Optimization Theory and Applications, Springer, vol. 197(2), pages 737-764, May.
    • Handle: RePEc:spr:joptap:v:197:y:2023:i:2:d:10.1007_s10957-023-02184-6
    DOI: 10.1007/s10957-023-02184-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-023-02184-6
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-023-02184-6?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. Hong-Zhi Wei & Chun-Rong Chen & Sheng-Jie Li, 2018. "Characterizations for Optimality Conditions of General Robust Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 177(3), pages 835-856, June.
    2. T. D. Chuong & V. H. Mak-Hau & J. Yearwood & R. Dazeley & M.-T. Nguyen & T. Cao, 2022. "Robust Pareto solutions for convex quadratic multiobjective optimization problems under data uncertainty," Annals of Operations Research, Springer, vol. 319(2), pages 1533-1564, December.
    3. Jiawei Chen & Elisabeth Köbis & Jen-Chih Yao, 2019. "Optimality Conditions and Duality for Robust Nonsmooth Multiobjective Optimization Problems with Constraints," Journal of Optimization Theory and Applications, Springer, vol. 181(2), pages 411-436, May.
    4. Thai Doan Chuong, 2022. "Second-order cone programming relaxations for a class of multiobjective convex polynomial problems," Annals of Operations Research, Springer, vol. 311(2), pages 1017-1033, April.
    5. Hong-Zhi Wei & Chun-Rong Chen & Sheng-Jie Li, 2020. "A Unified Approach Through Image Space Analysis to Robustness in Uncertain Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 184(2), pages 466-493, February.
    6. H. Yu & H. M. Liu, 2013. "Robust Multiple Objective Game Theory," Journal of Optimization Theory and Applications, Springer, vol. 159(1), pages 272-280, October.
    7. Jiawei Chen & Jun Li & Xiaobing Li & Yibing Lv & Jen-Chih Yao, 2020. "Radius of Robust Feasibility of System of Convex Inequalities with Uncertain Data," Journal of Optimization Theory and Applications, Springer, vol. 184(2), pages 384-399, February.
    8. Jiawang Nie, 2011. "Polynomial Matrix Inequality and Semidefinite Representation," Mathematics of Operations Research, INFORMS, vol. 36(3), pages 398-415, August.
    9. Xiangkai Sun & Kok Lay Teo & Xian-Jun Long, 2021. "Some Characterizations of Approximate Solutions for Robust Semi-infinite Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 191(1), pages 281-310, October.
    10. Radu Ioan Bot, 2010. "Conjugate Duality in Convex Optimization," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-642-04900-2, October.
    11. Cao Thanh Tinh & Thai Doan Chuong, 2022. "Conic Linear Programming Duals for Classes of Quadratic Semi-Infinite Programs with Applications," Journal of Optimization Theory and Applications, Springer, vol. 194(2), pages 570-596, August.
    12. Liguo Jiao & Jae Hyoung Lee, 2021. "Finding efficient solutions in robust multiple objective optimization with SOS-convex polynomial data," Annals of Operations Research, Springer, vol. 296(1), pages 803-820, January.
    Full references (including those not matched with items on IDEAS)

    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. Xiangkai Sun & Kok Lay Teo & Xian-Jun Long, 2021. "Some Characterizations of Approximate Solutions for Robust Semi-infinite Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 191(1), pages 281-310, October.
    2. S. Khoshkhabar-amiranloo, 2021. "Scalarization of Multiobjective Robust Optimization Problems," SN Operations Research Forum, Springer, vol. 2(3), pages 1-16, September.
    3. Siqin, Zhuoya & Niu, DongXiao & Li, MingYu & Gao, Tian & Lu, Yifan & Xu, Xiaomin, 2022. "Distributionally robust dispatching of multi-community integrated energy system considering energy sharing and profit allocation," Applied Energy, Elsevier, vol. 321(C).
    4. Hong-Zhi Wei & Chun-Rong Chen & Sheng-Jie Li, 2020. "Robustness Characterizations for Uncertain Optimization Problems via Image Space Analysis," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 459-479, August.
    5. 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.
    6. Zhe Hong & Kwan Deok Bae & Do Sang Kim, 2022. "Minimax programming as a tool for studying robust multi-objective optimization problems," Annals of Operations Research, Springer, vol. 319(2), pages 1589-1606, December.
    7. Jie Wang & Shengjie Li & Min Feng, 2022. "Unified Robust Necessary Optimality Conditions for Nonconvex Nonsmooth Uncertain Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 195(1), pages 226-248, October.
    8. Xiangkai Sun & Hongyong Fu & Jing Zeng, 2018. "Robust Approximate Optimality Conditions for Uncertain Nonsmooth Optimization with Infinite Number of Constraints," Mathematics, MDPI, vol. 7(1), pages 1-14, December.
    9. Feng Guo & Liguo Jiao, 2023. "A new scheme for approximating the weakly efficient solution set of vector rational optimization problems," Journal of Global Optimization, Springer, vol. 86(4), pages 905-930, August.
    10. Li Wang & Feng Guo, 2014. "Semidefinite relaxations for semi-infinite polynomial programming," Computational Optimization and Applications, Springer, vol. 58(1), pages 133-159, May.
    11. Vaithilingam Jeyakumar & Guoyin Li, 2017. "Exact Conic Programming Relaxations for a Class of Convex Polynomial Cone Programs," Journal of Optimization Theory and Applications, Springer, vol. 172(1), pages 156-178, January.
    12. Frauke Liers & Lars Schewe & Johannes Thürauf, 2022. "Radius of Robust Feasibility for Mixed-Integer Problems," INFORMS Journal on Computing, INFORMS, vol. 34(1), pages 243-261, January.
    13. T. D. Chuong & V. Jeyakumar & G. Li, 2019. "A new bounded degree hierarchy with SOCP relaxations for global polynomial optimization and conic convex semi-algebraic programs," Journal of Global Optimization, Springer, vol. 75(4), pages 885-919, December.
    14. Nguyen Minh Tung & Mai Duy, 2023. "Constraint qualifications and optimality conditions for robust nonsmooth semi-infinite multiobjective optimization problems," 4OR, Springer, vol. 21(1), pages 151-176, March.
    15. Jiawei Chen & Jun Li & Xiaobing Li & Yibing Lv & Jen-Chih Yao, 2020. "Radius of Robust Feasibility of System of Convex Inequalities with Uncertain Data," Journal of Optimization Theory and Applications, Springer, vol. 184(2), pages 384-399, February.
    16. Zhao, Jingang, 2018. "Three little-known and yet still significant contributions of Lloyd Shapley," Games and Economic Behavior, Elsevier, vol. 108(C), pages 592-599.
    17. Thai Doan Chuong & José Vicente-Pérez, 2023. "Conic Relaxations with Stable Exactness Conditions for Parametric Robust Convex Polynomial Problems," Journal of Optimization Theory and Applications, Springer, vol. 197(2), pages 387-410, May.
    18. Goberna, M.A. & Jeyakumar, V. & Li, G. & Vicente-Pérez, J., 2022. "The radius of robust feasibility of uncertain mathematical programs: A Survey and recent developments," European Journal of Operational Research, Elsevier, vol. 296(3), pages 749-763.
    19. Lu-Chuan Ceng & Xiaolong Qin & Yekini Shehu & Jen-Chih Yao, 2019. "Mildly Inertial Subgradient Extragradient Method for Variational Inequalities Involving an Asymptotically Nonexpansive and Finitely Many Nonexpansive Mappings," Mathematics, MDPI, vol. 7(10), pages 1-19, September.
    20. Ehrgott, Matthias & Ide, Jonas & Schöbel, Anita, 2014. "Minmax robustness for multi-objective optimization problems," European Journal of Operational Research, Elsevier, vol. 239(1), pages 17-31.

    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:197:y:2023:i:2:d:10.1007_s10957-023-02184-6. 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.