IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v76y2020i1d10.1007_s10898-019-00845-3.html
   My bibliography  Save this article

The decompositions with respect to two core non-symmetric cones

Author

Listed:
  • Yue Lu

    (Tianjin Normal University)

  • Ching-Yu Yang

    (National Taiwan Normal University)

  • Jein-Shan Chen

    (National Taiwan Normal University)

  • Hou-Duo Qi

    (University of Southampton)

Abstract

It is known that the analysis to tackle with non-symmetric cone optimization is quite different from the way to deal with symmetric cone optimization due to the discrepancy between these types of cones. However, there are still common concepts for both optimization problems, for example, the decomposition with respect to the given cone, smooth and nonsmooth analysis for the associated conic function, conic-convexity, conic-monotonicity and etc. In this paper, motivated by Chares’s thesis (Cones and interior-point algorithms for structured convex optimization involving powers and exponentials, 2009), we consider the decomposition issue of two core non-symmetric cones, in which two types of decomposition formulae will be proposed, one is adapted from the well-known Moreau decomposition theorem and the other follows from geometry properties of the given cones. As a byproduct, we also establish the conic functions of these cones and generalize the power cone case to its high-dimensional counterpart.

Suggested Citation

  • Yue Lu & Ching-Yu Yang & Jein-Shan Chen & Hou-Duo Qi, 2020. "The decompositions with respect to two core non-symmetric cones," Journal of Global Optimization, Springer, vol. 76(1), pages 155-188, January.
    • Handle: RePEc:spr:jglopt:v:76:y:2020:i:1:d:10.1007_s10898-019-00845-3
    DOI: 10.1007/s10898-019-00845-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-019-00845-3
    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/s10898-019-00845-3?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. Jein-Shan Chen, 2019. "SOC Functions and Their Applications," Springer Optimization and Its Applications, Springer, number 978-981-13-4077-2, September.
    2. F. Glineur & T. Terlaky, 2004. "Conic Formulation for l p -Norm Optimization," Journal of Optimization Theory and Applications, Springer, vol. 122(2), pages 285-307, August.
    3. Le Hien, 2015. "Differential properties of Euclidean projection onto power cone," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 82(3), pages 265-284, December.
    4. François Glineur, 2001. "Proving Strong Duality for Geometric Optimization Using a Conic Formulation," Annals of Operations Research, Springer, vol. 105(1), pages 155-184, July.
    5. Defeng Sun & Jie Sun, 2008. "Löwner's Operator and Spectral Functions in Euclidean Jordan Algebras," Mathematics of Operations Research, INFORMS, vol. 33(2), pages 421-445, May.
    6. Defeng Sun, 2006. "The Strong Second-Order Sufficient Condition and Constraint Nondegeneracy in Nonlinear Semidefinite Programming and Their Implications," Mathematics of Operations Research, INFORMS, vol. 31(4), pages 761-776, November.
    7. Jein-Shan Chen & Shaohua Pan, 2010. "An entropy-like proximal algorithm and the exponential multiplier method for convex symmetric cone programming," Computational Optimization and Applications, Springer, vol. 47(3), pages 477-499, November.
    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. Robert Chares & François Glineur, 2008. "An interior-point method for the single-facility location problem with mixed norms using a conic formulation," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 68(3), pages 383-405, December.
    2. F. Glineur & T. Terlaky, 2004. "Conic Formulation for l p -Norm Optimization," Journal of Optimization Theory and Applications, Springer, vol. 122(2), pages 285-307, August.
    3. Yingnan Wang & Naihua Xiu, 2011. "Strong Semismoothness of Projection onto Slices of Second-Order Cone," Journal of Optimization Theory and Applications, Springer, vol. 150(3), pages 599-614, September.
    4. Lingchen Kong & Levent Tunçel & Naihua Xiu, 2011. "Equivalent Conditions for Jacobian Nonsingularity in Linear Symmetric Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 148(2), pages 364-389, February.
    5. Liwei Zhang & Shengzhe Gao & Saoyan Guo, 2019. "Statistical Inference of Second-Order Cone Programming," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 36(02), pages 1-17, April.
    6. Shun Arahata & Takayuki Okuno & Akiko Takeda, 2023. "Complexity analysis of interior-point methods for second-order stationary points of nonlinear semidefinite optimization problems," Computational Optimization and Applications, Springer, vol. 86(2), pages 555-598, November.
    7. Qi Zhao & Zhongwen Chen, 2018. "An SQP-type Method with Superlinear Convergence for Nonlinear Semidefinite Programming," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 35(03), pages 1-25, June.
    8. Yong-Jin Liu & Li Wang, 2016. "Properties associated with the epigraph of the $$l_1$$ l 1 norm function of projection onto the nonnegative orthant," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(1), pages 205-221, August.
    9. M. V. Dolgopolik, 2018. "Augmented Lagrangian functions for cone constrained optimization: the existence of global saddle points and exact penalty property," Journal of Global Optimization, Springer, vol. 71(2), pages 237-296, June.
    10. Jingyong Tang & Hongchao Zhang, 2021. "A Nonmonotone Smoothing Newton Algorithm for Weighted Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 189(3), pages 679-715, June.
    11. Y. D. Chen & Y. Gao & Y.-J. Liu, 2010. "An Inexact SQP Newton Method for Convex SC1 Minimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 146(1), pages 33-49, July.
    12. Krokhmal, Pavlo A. & Soberanis, Policarpio, 2010. "Risk optimization with p-order conic constraints: A linear programming approach," European Journal of Operational Research, Elsevier, vol. 201(3), pages 653-671, March.
    13. Chengjin Li, 2014. "A New Approximation of the Matrix Rank Function and Its Application to Matrix Rank Minimization," Journal of Optimization Theory and Applications, Springer, vol. 163(2), pages 569-594, November.
    14. S. H. Pan & J.-S. Chen, 2009. "Growth Behavior of Two Classes of Merit Functions for Symmetric Cone Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 141(1), pages 167-191, April.
    15. Xiangjing Liu & Sanyang Liu, 2020. "A new nonmonotone smoothing Newton method for the symmetric cone complementarity problem with the Cartesian $$P_0$$ P 0 -property," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(2), pages 229-247, October.
    16. Qingna Li & Donghui Li & Houduo Qi, 2010. "Newton’s Method for Computing the Nearest Correlation Matrix with a Simple Upper Bound," Journal of Optimization Theory and Applications, Springer, vol. 147(3), pages 546-568, December.
    17. Huixian Wu & Hezhi Luo & Xiaodong Ding & Guanting Chen, 2013. "Global convergence of modified augmented Lagrangian methods for nonlinear semidefinite programming," Computational Optimization and Applications, Springer, vol. 56(3), pages 531-558, December.
    18. Shiyun Wang, 2020. "The Geometric Properties of a Class of Nonsymmetric Cones," Indian Journal of Pure and Applied Mathematics, Springer, vol. 51(3), pages 989-1002, September.
    19. Houduo Qi, 2009. "Local Duality of Nonlinear Semidefinite Programming," Mathematics of Operations Research, INFORMS, vol. 34(1), pages 124-141, February.
    20. H. Wu & H. Luo & J. Yang, 2014. "Nonlinear separation approach for the augmented Lagrangian in nonlinear semidefinite programming," Journal of Global Optimization, Springer, vol. 59(4), pages 695-727, August.

    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:76:y:2020:i:1:d:10.1007_s10898-019-00845-3. 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.