IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v75y2019i4d10.1007_s10898-019-00831-9.html
   My bibliography  Save this article

A new bounded degree hierarchy with SOCP relaxations for global polynomial optimization and conic convex semi-algebraic programs

Author

Listed:
  • T. D. Chuong

    (University of New South Wales
    Saigon University)

  • V. Jeyakumar

    (University of New South Wales)

  • G. Li

    (University of New South Wales)

Abstract

In this paper, we propose a bounded degree hierarchy of both primal and dual conic programming relaxations involving both semi-definite and second-order cone constraints for solving a nonconvex polynomial optimization problem with a bounded feasible set. This hierarchy makes use of some key aspects of the convergent linear programming relaxations of polynomial optimization problems (Lasserre in Moments, positive polynomials and their applications, World Scientific, Singapore, 2010) associated with Krivine–Stengle’s certificate of positivity in real algebraic geometry and some advantages of the scaled diagonally dominant sum of squares (SDSOS) polynomials (Ahmadi and Hall in Math Oper Res, 2019. https://doi.org/10.1287/moor.2018.0962; Ahmadi and Majumdar in SIAM J Appl Algebra Geom 3:193–230, 2019). We show that the values of both primal and dual relaxations converge to the global optimal value of the original polynomial optimization problem under some technical assumptions. Our hierarchy, which extends the so-called bounded degree Lasserre hierarchy (Lasserre et al. in Eur J Comput Optim 5:87–117, 2017), has a useful feature that the size and the number of the semi-definite and second-order cone constraints of the relaxations are fixed and independent of the step or level of the approximation in the hierarchy. As a special case, we provide a convergent bounded degree second-order cone programming (SOCP) hierarchy for solving polynomial optimization problems. We then present finite convergence at step one of the SOCP hierarchy for classes of polynomial optimization problems. This includes one-step convergence for a new class of first-order SDSOS-convex polynomial programs. In this case, we also show how a global solution is recovered from the level one SOCP relaxation. We finally derive a corresponding convergent conic linear programming hierarchy for conic-convex semi-algebraic programs. Whenever the semi-algebraic set of the conic-convex program is described by concave polynomial inequalities, we show further that the values of the relaxation problems converge to the common value of the convex program and its Lagrangian dual under a constraint qualification.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:jglopt:v:75:y:2019:i:4:d:10.1007_s10898-019-00831-9
    DOI: 10.1007/s10898-019-00831-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-019-00831-9
    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/s10898-019-00831-9?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. Jiawang Nie, 2011. "Polynomial Matrix Inequality and Semidefinite Representation," Mathematics of Operations Research, INFORMS, vol. 36(3), pages 398-415, August.
    2. Shenglong Hu & Guoyin Li & Liqun Qi, 2016. "A Tensor Analogy of Yuan’s Theorem of the Alternative and Polynomial Optimization with Sign structure," Journal of Optimization Theory and Applications, Springer, vol. 168(2), pages 446-474, February.
    3. T. D. Chuong & V. Jeyakumar, 2018. "Generalized Lagrangian duality for nonconvex polynomial programs with polynomial multipliers," Journal of Global Optimization, Springer, vol. 72(4), pages 655-678, December.
    4. Jean B. Lasserre & Kim-Chuan Toh & Shouguang Yang, 2017. "A bounded degree SOS hierarchy for polynomial optimization," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 5(1), pages 87-117, March.
    5. Xiaolong Kuang & Bissan Ghaddar & Joe Naoum-Sawaya & Luis F. Zuluaga, 2019. "Alternative SDP and SOCP approximations for polynomial optimization," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 7(2), pages 153-175, June.
    6. 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.
    7. V. Jeyakumar, 2008. "Constraint Qualifications Characterizing Lagrangian Duality in Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 136(1), pages 31-41, January.
    8. Laurent, M., 2009. "Sums of squares, moment matrices and optimization over polynomials," Other publications TiSEM 9fef820b-69d2-43f2-a501-e, Tilburg University, School of Economics and Management.
    9. Amir Ali Ahmadi & Georgina Hall, 2019. "On the Construction of Converging Hierarchies for Polynomial Optimization Based on Certificates of Global Positivity," Management Science, INFORMS, vol. 44(4), pages 1192-1207, November.
    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. Meng-Meng Zheng & Zheng-Hai Huang & Sheng-Long Hu, 2022. "Unconstrained minimization of block-circulant polynomials via semidefinite program in third-order tensor space," Journal of Global Optimization, Springer, vol. 84(2), pages 415-440, October.
    2. T. D. Chuong & V. Jeyakumar & G. Li & D. Woolnough, 2021. "Exact SDP reformulations of adjustable robust linear programs with box uncertainties under separable quadratic decision rules via SOS representations of non-negativity," Journal of Global Optimization, Springer, vol. 81(4), pages 1095-1117, December.
    3. Xiangkai Sun & Jiayi Huang & Kok Lay Teo, 2024. "On semidefinite programming relaxations for a class of robust SOS-convex polynomial optimization problems," Journal of Global Optimization, Springer, vol. 88(3), pages 755-776, March.
    4. Thai Doan Chuong & Xinghuo Yu & Andrew Eberhard & Chaojie Li & Chen Liu, 2024. "Hierarchy relaxations for robust equilibrium constrained polynomial problems and applications to electric vehicle charging scheduling," Journal of Global Optimization, Springer, vol. 90(3), pages 781-811, November.
    5. Cao Thanh Tinh & Thai Doan Chuong, 2024. "Robust second order cone conditions and duality for multiobjective problems under uncertainty data," Journal of Global Optimization, Springer, vol. 88(4), pages 901-926, April.
    6. 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.
    7. 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.
    8. 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.

    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. Meng-Meng Zheng & Zheng-Hai Huang & Sheng-Long Hu, 2022. "Unconstrained minimization of block-circulant polynomials via semidefinite program in third-order tensor space," Journal of Global Optimization, Springer, vol. 84(2), pages 415-440, October.
    2. Thai Doan Chuong & José Vicente-Pérez, 2025. "Conic relaxations for conic minimax convex polynomial programs with extensions and applications," Journal of Global Optimization, Springer, vol. 91(4), pages 743-763, April.
    3. Peter J. C. Dickinson & Janez Povh, 2019. "A new approximation hierarchy for polynomial conic optimization," Computational Optimization and Applications, Springer, vol. 73(1), pages 37-67, May.
    4. Sönke Behrends & Anita Schöbel, 2020. "Generating Valid Linear Inequalities for Nonlinear Programs via Sums of Squares," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 911-935, September.
    5. Moslem Zamani, 2019. "A new algorithm for concave quadratic programming," Journal of Global Optimization, Springer, vol. 75(3), pages 655-681, November.
    6. Xiao Wang & Xinzhen Zhang & Guangming Zhou, 2020. "SDP relaxation algorithms for $$\mathbf {P}(\mathbf {P}_0)$$P(P0)-tensor detection," Computational Optimization and Applications, Springer, vol. 75(3), pages 739-752, April.
    7. Laurent, Monique & Vargas, Luis Felipe, 2022. "Finite convergence of sum-of-squares hierarchies for the stability number of a graph," Other publications TiSEM 3998b864-7504-4cf4-bc1d-f, Tilburg University, School of Economics and Management.
    8. Fabián Flores-Bazán & William Echegaray & Fernando Flores-Bazán & Eladio Ocaña, 2017. "Primal or dual strong-duality in nonconvex optimization and a class of quasiconvex problems having zero duality gap," Journal of Global Optimization, Springer, vol. 69(4), pages 823-845, December.
    9. Polyxeni-Margarita Kleniati & Panos Parpas & Berç Rustem, 2010. "Partitioning procedure for polynomial optimization," Journal of Global Optimization, Springer, vol. 48(4), pages 549-567, December.
    10. Laurent, M. & Rostalski, P., 2012. "The approach of moments for polynomial equations," Other publications TiSEM f08f3cd2-b83e-4bf1-9322-a, Tilburg University, School of Economics and Management.
    11. Jie Wang & Victor Magron, 2021. "Exploiting term sparsity in noncommutative polynomial optimization," Computational Optimization and Applications, Springer, vol. 80(2), pages 483-521, November.
    12. Tomohiko Mizutani & Makoto Yamashita, 2013. "Correlative sparsity structures and semidefinite relaxations for concave cost transportation problems with change of variables," Journal of Global Optimization, Springer, vol. 56(3), pages 1073-1100, July.
    13. Li Wang & Feng Guo, 2014. "Semidefinite relaxations for semi-infinite polynomial programming," Computational Optimization and Applications, Springer, vol. 58(1), pages 133-159, May.
    14. Fook Wai Kong & Polyxeni-Margarita Kleniati & Berç Rustem, 2012. "Computation of Correlated Equilibrium with Global-Optimal Expected Social Welfare," Journal of Optimization Theory and Applications, Springer, vol. 153(1), pages 237-261, April.
    15. de Klerk, E. & Laurent, M., 2010. "Error bounds for some semidefinite programming approaches to polynomial minimization on the hypercube," Other publications TiSEM 619d9658-77df-4b5e-9868-0, Tilburg University, School of Economics and Management.
    16. Sandra S. Y. Tan & Antonios Varvitsiotis & Vincent Y. F. Tan, 2021. "Analysis of Optimization Algorithms via Sum-of-Squares," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 56-81, July.
    17. Guanglei Wang & Hassan Hijazi, 2018. "Mathematical programming methods for microgrid design and operations: a survey on deterministic and stochastic approaches," Computational Optimization and Applications, Springer, vol. 71(2), pages 553-608, November.
    18. Xiaolong Kuang & Bissan Ghaddar & Joe Naoum-Sawaya & Luis F. Zuluaga, 2019. "Alternative SDP and SOCP approximations for polynomial optimization," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 7(2), pages 153-175, June.
    19. Campos, Juan S. & Misener, Ruth & Parpas, Panos, 2019. "A multilevel analysis of the Lasserre hierarchy," European Journal of Operational Research, Elsevier, vol. 277(1), pages 32-41.
    20. Ahmadreza Marandi & Joachim Dahl & Etienne Klerk, 2018. "A numerical evaluation of the bounded degree sum-of-squares hierarchy of Lasserre, Toh, and Yang on the pooling problem," Annals of Operations Research, Springer, vol. 265(1), pages 67-92, June.

    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:75:y:2019:i:4:d:10.1007_s10898-019-00831-9. 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.