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On Chebyshev Center of the Intersection of Two Ellipsoids

  • Xiaoli Cen
  • Yong XiaEmail author
  • Runxuan Gao
  • Tianzhi Yang
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 991)

Abstract

We study the problem of finding the smallest ball covering the intersection of two ellipsoids, which is also known as the Chebyshev center problem (CC). Semidefinite programming (SDP) relaxation is an efficient approach to approximate (CC). In this paper, we first establish the worst-case approximation bound of (SDP). Then we show that (CC) can be globally solved in polynomial time. As a by-product, one can randomly generate Celis-Dennis-Tapia subproblems having positive Lagrangian duality gap with high probability.

Keywords

Chebyshev center Semidefinite programming Approximation bound Polynomial solvability CDT subproblem 

Notes

Acknowledgments

This research was supported by National Natural Science Foundation of China under grants 11822103, 11571029, 11771056 and Beijing Natural Science Foundation Z180005.

References

  1. 1.
    Ai, W., Zhang, S.: Strong duality for the CDT subproblem: a necessary and sufficient condition. SIAM J. Optim. 19(4), 1735–1756 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Beck, A.: Convexity properties associated with nonconvex quadratic matrix functions and applications to quadratic programming. J. Optim. Theory Appl. 142(1), 1–29 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Beck, A., Eldar, Y.: Strong duality in nonconvex quadratic optimization with two quadratic constraints. SIAM J. Optim. 17(3), 844–860 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Beck, A., Eldar, Y.: Regularization in regression with bounded noise: a Chebyshev center approach. SIAM J. Matrix Anal. Appl. 29(2), 606–625 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bienstock, D.: A note on polynomial solvability of the CDT problem. SIAM J. Optim. 26(1), 488–498 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Burer, S.: A gentle, geometric introduction to copositive optimization. Math. Program. 151(1), 89–116 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Burer, S., Anstreicher, K.M.: Second-order-cone constraints for extended trust-region subproblems. SIAM J. Optim. 23(1), 432–451 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chen, X., Yuan, Y.: On local solutions of the Celis-Dennis-Tapia subproblem. SIAM J. Optim. 10(2), 359–383 (2000)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Consolini, L., Locatelli, M.: On the complexity of quadratic programming with two quadratic constraints. Math. Program. 164(1–2), 91–128 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Eldar, Y., Beck, A.: A minimax Chebyshev estimator for bounded error estimation. IEEE Trans. Signal Process. 56(4), 1388–1397 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming error estimation, version 2.1. (March 2014). http://cvxr.com/cvx
  12. 12.
    Hsia, Y., Wang, S., Xu, Z.: Improved semidefinite approximation bounds for nonconvex nonhomogeneous quadratic optimization with ellipsoid constraints. Oper. Res. Lett. 43(4), 378–383 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Milanese, M., Vicino, A.: Optimal estimation theory for dynamic systems with set membership uncertainty: an overview. Automatica 27(6), 997–1009 (1991)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Nesterov, Y.: Introductory Lectures on Convex Optimizaiton: A Basic Course. Kluwer Academic, Boston (2004)CrossRefGoogle Scholar
  15. 15.
    Sakaue, S., Nakatsukasa, Y., Takeda, A., Iwata, S.: Solving generalized CDT problems via two-parameter eigenvalues. SIAM J. Optim. 26(3), 1669–1694 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Xia, Y., Yang, M., Wang, S.: Chebyshev center of the intersection of balls: complexity, relaxation and approximation (2019). arXiv:1901.07645
  17. 17.
    Yang, B., Burer, S.: A two-variable approach to the two-trust-region subproblem. SIAM J. Optim. 26(1), 661–680 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ye, Y., Zhang, S.: New results on quadratic minimization. SIAM J. Optim. 14(1), 245–267 (2003)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Yuan, J., Wang, M., Ai, W., Shuai, T.: New results on narrowing the duality gap of the extended Celis-Dennis-Tapia problem. SIAM J. Optim. 27(2), 890–909 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Yuan, Y.: On a subproblem of trust region algorithms for constrained optimization. Math. Program. 47(1–3), 53–63 (1990)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Xiaoli Cen
    • 1
  • Yong Xia
    • 1
    Email author
  • Runxuan Gao
    • 1
  • Tianzhi Yang
    • 1
  1. 1.LMIB of the Ministry of Education; School of Mathematics and System SciencesBeihang UniversityBeijingPeople’s Republic of China

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