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.
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References
Ai, W., Zhang, S.: Strong duality for the CDT subproblem: a necessary and sufficient condition. SIAM J. Optim. 19(4), 1735–1756 (2009)
Beck, A.: Convexity properties associated with nonconvex quadratic matrix functions and applications to quadratic programming. J. Optim. Theory Appl. 142(1), 1–29 (2009)
Beck, A., Eldar, Y.: Strong duality in nonconvex quadratic optimization with two quadratic constraints. SIAM J. Optim. 17(3), 844–860 (2006)
Beck, A., Eldar, Y.: Regularization in regression with bounded noise: a Chebyshev center approach. SIAM J. Matrix Anal. Appl. 29(2), 606–625 (2007)
Bienstock, D.: A note on polynomial solvability of the CDT problem. SIAM J. Optim. 26(1), 488–498 (2016)
Burer, S.: A gentle, geometric introduction to copositive optimization. Math. Program. 151(1), 89–116 (2015)
Burer, S., Anstreicher, K.M.: Second-order-cone constraints for extended trust-region subproblems. SIAM J. Optim. 23(1), 432–451 (2013)
Chen, X., Yuan, Y.: On local solutions of the Celis-Dennis-Tapia subproblem. SIAM J. Optim. 10(2), 359–383 (2000)
Consolini, L., Locatelli, M.: On the complexity of quadratic programming with two quadratic constraints. Math. Program. 164(1–2), 91–128 (2017)
Eldar, Y., Beck, A.: A minimax Chebyshev estimator for bounded error estimation. IEEE Trans. Signal Process. 56(4), 1388–1397 (2008)
Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming error estimation, version 2.1. (March 2014). http://cvxr.com/cvx
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)
Milanese, M., Vicino, A.: Optimal estimation theory for dynamic systems with set membership uncertainty: an overview. Automatica 27(6), 997–1009 (1991)
Nesterov, Y.: Introductory Lectures on Convex Optimizaiton: A Basic Course. Kluwer Academic, Boston (2004)
Sakaue, S., Nakatsukasa, Y., Takeda, A., Iwata, S.: Solving generalized CDT problems via two-parameter eigenvalues. SIAM J. Optim. 26(3), 1669–1694 (2016)
Xia, Y., Yang, M., Wang, S.: Chebyshev center of the intersection of balls: complexity, relaxation and approximation (2019). arXiv:1901.07645
Yang, B., Burer, S.: A two-variable approach to the two-trust-region subproblem. SIAM J. Optim. 26(1), 661–680 (2016)
Ye, Y., Zhang, S.: New results on quadratic minimization. SIAM J. Optim. 14(1), 245–267 (2003)
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)
Yuan, Y.: On a subproblem of trust region algorithms for constrained optimization. Math. Program. 47(1–3), 53–63 (1990)
Acknowledgments
This research was supported by National Natural Science Foundation of China under grants 11822103, 11571029, 11771056 and Beijing Natural Science Foundation Z180005.
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Cen, X., Xia, Y., Gao, R., Yang, T. (2020). On Chebyshev Center of the Intersection of Two Ellipsoids. In: Le Thi, H., Le, H., Pham Dinh, T. (eds) Optimization of Complex Systems: Theory, Models, Algorithms and Applications. WCGO 2019. Advances in Intelligent Systems and Computing, vol 991. Springer, Cham. https://doi.org/10.1007/978-3-030-21803-4_14
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DOI: https://doi.org/10.1007/978-3-030-21803-4_14
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