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LMI Approximations for the Radius of the Intersection of Ellipsoids: Survey

Author

Listed:
  • D. Henrion

    (Centre National de la Recherche Scientifique)

  • S. Tarbouriech

    (Centre National de la Recherche Scientifique)

  • D. Arzelier

    (Centre National de la Recherche Scientifique)

Abstract

This paper surveys various linear matrix inequality relaxation techniques for evaluating the maximum norm vector within the intersection of several ellipsoids. This difficult nonconvex optimization problem arises frequently in robust control synthesis. Two randomized algorithms and several ellipsoidal approximations are described. Guaranteed approximation bounds are derived in order to evaluate the quality of these relaxations.

Suggested Citation

  • D. Henrion & S. Tarbouriech & D. Arzelier, 2001. "LMI Approximations for the Radius of the Intersection of Ellipsoids: Survey," Journal of Optimization Theory and Applications, Springer, vol. 108(1), pages 1-28, January.
    • Handle: RePEc:spr:joptap:v:108:y:2001:i:1:d:10.1023_a:1026454804250
    DOI: 10.1023/A:1026454804250
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    References listed on IDEAS

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    1. Minyue Fu & Zhi-Quan Luo & Yinyu Ye, 1998. "Approximation Algorithms for Quadratic Programming," Journal of Combinatorial Optimization, Springer, vol. 2(1), pages 29-50, March.
    2. NESTEROV, Yu., 1998. "Semidefinite relaxation and nonconvex quadratic optimization," LIDAM Reprints CORE 1362, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. A. Ben-Tal & A. Nemirovski, 1998. "Robust Convex Optimization," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 769-805, November.
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    Cited by:

    1. Zhuoyi Xu & Linbin Li & Yong Xia, 2023. "A partial ellipsoidal approximation scheme for nonconvex homogeneous quadratic optimization with quadratic constraints," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 98(1), pages 93-109, August.
    2. Tongli Zhang & Yong Xia, 2022. "Comment on “Approximation algorithms for quadratic programming”," Journal of Combinatorial Optimization, Springer, vol. 44(2), pages 1099-1103, September.
    3. Andreas Rauh & Stefan Wirtensohn & Patrick Hoher & Johannes Reuter & Luc Jaulin, 2022. "Reliability Assessment of an Unscented Kalman Filter by Using Ellipsoidal Enclosure Techniques," Mathematics, MDPI, vol. 10(16), pages 1-18, August.

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