IDEAS home Printed from https://ideas.repec.org/a/wly/jnlaaa/v2012y2012i1n836804.html

A Geometric Mean of Parameterized Arithmetic and Harmonic Means of Convex Functions

Author

Listed:
  • Sangho Kum
  • Yongdo Lim

Abstract

The notion of the geometric mean of two positive reals is extended by Ando (1978) to the case of positive semidefinite matrices A and B. Moreover, an interesting generalization of the geometric mean A # B of A and B to convex functions was introduced by Atteia and Raïssouli (2001) with a different viewpoint of convex analysis. The present work aims at providing a further development of the geometric mean of convex functions due to Atteia and Raïssouli (2001). A new algorithmic self‐dual operator for convex functions named “the geometric mean of parameterized arithmetic and harmonic means of convex functions” is proposed, and its essential properties are investigated.

Suggested Citation

  • Sangho Kum & Yongdo Lim, 2012. "A Geometric Mean of Parameterized Arithmetic and Harmonic Means of Convex Functions," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    • Handle: RePEc:wly:jnlaaa:v:2012:y:2012:i:1:n:836804
    DOI: 10.1155/2012/836804
    as

    Download full text from publisher

    File URL: https://doi.org/10.1155/2012/836804
    Download Restriction: no
    File URL: https://libkey.io/10.1155/2012/836804?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
    ---><---

    References listed on IDEAS

    as
    1. Yu. E. Nesterov & M. J. Todd, 1997. "Self-Scaled Barriers and Interior-Point Methods for Convex Programming," Mathematics of Operations Research, INFORMS, vol. 22(1), pages 1-42, February.
    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. Chee-Khian Sim, 2019. "Interior point method on semi-definite linear complementarity problems using the Nesterov–Todd (NT) search direction: polynomial complexity and local convergence," Computational Optimization and Applications, Springer, vol. 74(2), pages 583-621, November.
    2. Sturm, J.F., 2001. "Avoiding Numerical Cancellation in the Interior Point Method for Solving Semidefinite Programs," Other publications TiSEM 949fb20a-a2c6-4d87-85ea-8, Tilburg University, School of Economics and Management.
    3. B.V. Halldórsson & R.H. Tütüncü, 2003. "An Interior-Point Method for a Class of Saddle-Point Problems," Journal of Optimization Theory and Applications, Springer, vol. 116(3), pages 559-590, March.
    4. G. Q. Wang & Y. Q. Bai, 2012. "A New Full Nesterov–Todd Step Primal–Dual Path-Following Interior-Point Algorithm for Symmetric Optimization," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 966-985, September.
    5. G. Q. Wang & L. C. Kong & J. Y. Tao & G. Lesaja, 2015. "Improved Complexity Analysis of Full Nesterov–Todd Step Feasible Interior-Point Method for Symmetric Optimization," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 588-604, August.
    6. E. A. Yıldırım, 2003. "An Interior-Point Perspective on Sensitivity Analysis in Semidefinite Programming," Mathematics of Operations Research, INFORMS, vol. 28(4), pages 649-676, November.
    7. Vasile L. Basescu & John E. Mitchell, 2008. "An Analytic Center Cutting Plane Approach for Conic Programming," Mathematics of Operations Research, INFORMS, vol. 33(3), pages 529-551, August.
    8. G. Q. Wang & Y. Q. Bai & X. Y. Gao & D. Z. Wang, 2015. "Improved Complexity Analysis of Full Nesterov–Todd Step Interior-Point Methods for Semidefinite Optimization," Journal of Optimization Theory and Applications, Springer, vol. 165(1), pages 242-262, April.
    9. J.F. Sturm & S. Zhang, 1998. "On Sensitivity of Central Solutions in Semidefinite Programming," Tinbergen Institute Discussion Papers 98-040/4, Tinbergen Institute.
    10. Mehdi Karimi & Levent Tunçel, 2020. "Primal–Dual Interior-Point Methods for Domain-Driven Formulations," Mathematics of Operations Research, INFORMS, vol. 45(2), pages 591-621, May.
    11. Changhe Liu & Hongwei Liu, 2012. "A new second-order corrector interior-point algorithm for semidefinite programming," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 75(2), pages 165-183, April.
    12. Behrouz Kheirfam, 2015. "A Corrector–Predictor Path-Following Method for Convex Quadratic Symmetric Cone Optimization," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 246-260, January.
    13. Mehdi Karimi & Levent Tunçel, 2025. "Efficient Implementation of Interior-Point Methods for Quantum Relative Entropy," INFORMS Journal on Computing, INFORMS, vol. 37(1), pages 3-21, January.
    14. Renato D. C. Monteiro & Paulo R. Zanjácomo, 2000. "General Interior-Point Maps and Existence of Weighted Paths for Nonlinear Semidefinite Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 25(3), pages 381-399, August.
    15. Quoc Tran-Dinh & Anastasios Kyrillidis & Volkan Cevher, 2018. "A Single-Phase, Proximal Path-Following Framework," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1326-1347, November.
    16. Zsolt Darvay & Petra Renáta Rigó, 2024. "New Predictor–Corrector Algorithm for Symmetric Cone Horizontal Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 202(1), pages 50-75, July.
    17. Sturm, J.F., 2001. "Avoiding Numerical Cancellation in the Interior Point Method for Solving Semidefinite Programs," Discussion Paper 2001-27, Tilburg University, Center for Economic Research.
    18. Zhaonan Qu & Wenzhi Gao & Oliver Hinder & Yinyu Ye & Zhengyuan Zhou, 2025. "Optimal Diagonal Preconditioning," Operations Research, INFORMS, vol. 73(3), pages 1479-1495, May.
    19. Sungwoo Park & Dianne P. O’Leary, 2015. "A Polynomial Time Constraint-Reduced Algorithm for Semidefinite Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 558-571, August.
    20. Peng, Jiming & Roos, Cornelis & Terlaky, Tamas, 2002. "A new class of polynomial primal-dual methods for linear and semidefinite optimization," European Journal of Operational Research, Elsevier, vol. 143(2), pages 234-256, December.

    More about this item

    Statistics

    Access and download statistics

    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:wly:jnlaaa:v:2012:y:2012:i:1:n:836804. 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: Wiley Content Delivery (email available below). General contact details of provider: https://onlinelibrary.wiley.com/journal/4058 .

    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.