IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v80y2021i1d10.1007_s10589-021-00289-0.html
   My bibliography  Save this article

An efficient global algorithm for worst-case linear optimization under uncertainties based on nonlinear semidefinite relaxation

Author

Listed:
  • Xiaodong Ding

    (Zhejiang University of Technology)

  • Hezhi Luo

    (Zhejiang Sci-Tech University)

  • Huixian Wu

    (Hangzhou Dianzi University)

  • Jianzhen Liu

    (Hangzhou Dianzi University)

Abstract

The worst-case linear optimization (WCLO) with uncertainties in the right-hand-side of the constraints often arises from numerous applications such as systemic risk estimate in finance and stochastic optimization, which is known to be NP-hard. In this paper, we investigate the efficient global algorithm for WCLO based on its nonlinear semidefinite relaxation (SDR). We first derive an enhanced nonlinear SDR for WCLO via secant cuts and RLT approaches. A secant search algorithm is then proposed to solve the nonlinear SDR and its global convergence is established. Second, we propose a new global algorithm for WCLO, which integrates the nonlinear SDR with successive convex optimization method, initialization and branch-and-bound, to find a globally optimal solution to the underlying WCLO within a pre-specified $$\epsilon$$ ϵ -tolerance. We establish the global convergence of the algorithm and estimate its complexity. Preliminary numerical results demonstrate that the proposed algorithm can effectively find a globally optimal solution to the WCLO instances.

Suggested Citation

  • Xiaodong Ding & Hezhi Luo & Huixian Wu & Jianzhen Liu, 2021. "An efficient global algorithm for worst-case linear optimization under uncertainties based on nonlinear semidefinite relaxation," Computational Optimization and Applications, Springer, vol. 80(1), pages 89-120, September.
    • Handle: RePEc:spr:coopap:v:80:y:2021:i:1:d:10.1007_s10589-021-00289-0
    DOI: 10.1007/s10589-021-00289-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-021-00289-0
    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/s10589-021-00289-0?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. NESTEROV, Yu., 1998. "Semidefinite relaxation and nonconvex quadratic optimization," LIDAM Reprints CORE 1362, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. 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.
    3. A. Ben-Tal & A. Nemirovski, 1998. "Robust Convex Optimization," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 769-805, November.
    4. Jiming Peng & Tao Zhu & Hezhi Luo & Kim-Chuan Toh, 2015. "Semi-definite programming relaxation of quadratic assignment problems based on nonredundant matrix splitting," Computational Optimization and Applications, Springer, vol. 60(1), pages 171-198, January.
    5. Larry Eisenberg & Thomas H. Noe, 2001. "Systemic Risk in Financial Systems," Management Science, INFORMS, vol. 47(2), pages 236-249, February.
    6. Hezhi Luo & Xiaodi Bai & Jiming Peng, 2019. "Enhancing Semidefinite Relaxation for Quadratically Constrained Quadratic Programming via Penalty Methods," Journal of Optimization Theory and Applications, Springer, vol. 180(3), pages 964-992, March.
    7. Xin Chen & Melvyn Sim & Peng Sun, 2007. "A Robust Optimization Perspective on Stochastic Programming," Operations Research, INFORMS, vol. 55(6), pages 1058-1071, December.
    8. Samuel Burer & Dieter Vandenbussche, 2009. "Globally solving box-constrained nonconvex quadratic programs with semidefinite-based finite branch-and-bound," Computational Optimization and Applications, Springer, vol. 43(2), pages 181-195, June.
    9. Dimitris Bertsimas & Vineet Goyal, 2010. "On the Power of Robust Solutions in Two-Stage Stochastic and Adaptive Optimization Problems," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 284-305, May.
    10. X. Zheng & X. Sun & D. Li, 2011. "Nonconvex quadratically constrained quadratic programming: best D.C. decompositions and their SDP representations," Journal of Global Optimization, Springer, vol. 50(4), pages 695-712, 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. Huixian Wu & Hezhi Luo & Xianye Zhang & Jianzhen Liu, 2023. "A new global algorithm for factor-risk-constrained mean-variance portfolio selection," Journal of Global Optimization, Springer, vol. 87(2), pages 503-532, November.
    2. Huixian Wu & Hezhi Luo & Xianye Zhang & Haiqiang Qi, 2023. "An effective global algorithm for worst-case linear optimization under polyhedral uncertainty," Journal of Global Optimization, Springer, vol. 87(1), pages 191-219, September.
    3. Hezhi Luo & Xianye Zhang & Huixian Wu & Weiqiang Xu, 2023. "Effective algorithms for separable nonconvex quadratic programming with one quadratic and box constraints," Computational Optimization and Applications, Springer, vol. 86(1), pages 199-240, 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. 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.
    2. Huixian Wu & Hezhi Luo & Xianye Zhang & Haiqiang Qi, 2023. "An effective global algorithm for worst-case linear optimization under polyhedral uncertainty," Journal of Global Optimization, Springer, vol. 87(1), pages 191-219, September.
    3. Wenqing Chen & Melvyn Sim & Jie Sun & Chung-Piaw Teo, 2010. "From CVaR to Uncertainty Set: Implications in Joint Chance-Constrained Optimization," Operations Research, INFORMS, vol. 58(2), pages 470-485, April.
    4. Hamed Mamani & Shima Nassiri & Michael R. Wagner, 2017. "Closed-Form Solutions for Robust Inventory Management," Management Science, INFORMS, vol. 63(5), pages 1625-1643, May.
    5. L. Jeff Hong & Zhiyuan Huang & Henry Lam, 2021. "Learning-Based Robust Optimization: Procedures and Statistical Guarantees," Management Science, INFORMS, vol. 67(6), pages 3447-3467, June.
    6. Fanwen Meng & Jin Qi & Meilin Zhang & James Ang & Singfat Chu & Melvyn Sim, 2015. "A Robust Optimization Model for Managing Elective Admission in a Public Hospital," Operations Research, INFORMS, vol. 63(6), pages 1452-1467, December.
    7. Wenqing Chen & Melvyn Sim, 2009. "Goal-Driven Optimization," Operations Research, INFORMS, vol. 57(2), pages 342-357, April.
    8. Hezhi Luo & Yuanyuan Chen & Xianye Zhang & Duan Li & Huixian Wu, 2020. "Effective Algorithms for Optimal Portfolio Deleveraging Problem with Cross Impact," Papers 2012.07368, arXiv.org, revised Jan 2021.
    9. Postek, Krzysztof & Ben-Tal, A. & den Hertog, Dick & Melenberg, Bertrand, 2015. "Exact Robust Counterparts of Ambiguous Stochastic Constraints Under Mean and Dispersion Information," Other publications TiSEM d718e419-a375-4707-b206-e, Tilburg University, School of Economics and Management.
    10. 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.
    11. Dimitris Bertsimas & Vineet Goyal, 2013. "On the approximability of adjustable robust convex optimization under uncertainty," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(3), pages 323-343, June.
    12. An, Kun & Lo, Hong K., 2016. "Two-phase stochastic program for transit network design under demand uncertainty," Transportation Research Part B: Methodological, Elsevier, vol. 84(C), pages 157-181.
    13. Postek, K.S. & den Hertog, D., 2016. "Multi-stage Adjustable Robust Mixed-Integer Optimization via Iterative Splitting of the Uncertainty set (Revision of CentER Discussion Paper 2014-056)," Discussion Paper 2016-006, Tilburg University, Center for Economic Research.
    14. Karthik Natarajan & Dessislava Pachamanova & Melvyn Sim, 2008. "Incorporating Asymmetric Distributional Information in Robust Value-at-Risk Optimization," Management Science, INFORMS, vol. 54(3), pages 573-585, March.
    15. Joel Goh & Melvyn Sim, 2011. "Robust Optimization Made Easy with ROME," Operations Research, INFORMS, vol. 59(4), pages 973-985, August.
    16. Shipra Agrawal & Yichuan Ding & Amin Saberi & Yinyu Ye, 2012. "Price of Correlations in Stochastic Optimization," Operations Research, INFORMS, vol. 60(1), pages 150-162, February.
    17. Postek, K.S. & den Hertog, D., 2014. "Multi-stage Adjustable Robust Mixed-Integer Optimization via Iterative Splitting of the Uncertainty set," Other publications TiSEM 8649012b-1bb0-4d34-83f1-7, Tilburg University, School of Economics and Management.
    18. Gabrel, Virginie & Murat, Cécile & Thiele, Aurélie, 2014. "Recent advances in robust optimization: An overview," European Journal of Operational Research, Elsevier, vol. 235(3), pages 471-483.
    19. Long He & Ho-Yin Mak & Ying Rong & Zuo-Jun Max Shen, 2017. "Service Region Design for Urban Electric Vehicle Sharing Systems," Manufacturing & Service Operations Management, INFORMS, vol. 19(2), pages 309-327, May.
    20. Nalan Gülpınar & Dessislava Pachamanova & Ethem Çanakoğlu, 2016. "A robust asset–liability management framework for investment products with guarantees," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 38(4), pages 1007-1041, October.

    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:80:y:2021:i:1:d:10.1007_s10589-021-00289-0. 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.