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On Conic Relaxations of Generalization of the Extended Trust Region Subproblem

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Optimization of Complex Systems: Theory, Models, Algorithms and Applications (WCGO 2019)

Abstract

The extended trust region subproblem (ETRS) of minimizing a quadratic objective over the unit ball with additional linear constraints has attracted a lot of attention in the last few years due to its theoretical significance and wide spectra of applications. Several sufficient conditions to guarantee the exactness of its semidefinite programming (SDP) relaxation or second order cone programming (SOCP) relaxation have been recently developed in the literature. In this paper, we consider a generalization of the extended trust region subproblem (GETRS), in which the unit ball constraint in ETRS is replaced by a general, possibly nonconvex, quadratic constraint. We demonstrate that the SDP relaxation can further be reformulated as an SOCP problem under a simultaneous diagonalization condition of the quadratic form. We then explore several sufficient conditions under which the SOCP relaxation of GETRS is exact under Slater condition.

Supported by Shanghai Sailing Program 18YF1401700, Natural Science Foundation of China (NSFC) 11801087 and Hong Kong Research Grants Council under Grants 14213716 and 14202017.

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References

  1. Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust optimization. Princeton University Press (2009)

    Google Scholar 

  2. Ben-Tal, A., den Hertog, D.: Hidden conic quadratic representation of some nonconvex quadratic optimization problems. Math. Program. 143(1–2), 1–29 (2014)

    Google Scholar 

  3. Ben-Tal, A., Teboulle, M.: Hidden convexity in some nonconvex quadratically constrained quadratic programming. Math. Program. 72(1), 51–63 (1996)

    Google Scholar 

  4. Boyd, S., Vandenberghe, L.: Convex optimization. Cambridge University Press (2004)

    Google Scholar 

  5. Burer, S., Anstreicher, K.M.: Second-order-cone constraints for extended trust-region subproblems. SIAM J. Optim. 23(1), 432–451 (2013)

    Google Scholar 

  6. Burer, S., Yang, B.: The trust region subproblem with non-intersecting linear constraints. Math. Program. 149(1–2), 253–264 (2015)

    Google Scholar 

  7. Conn, A.R., Gould, N.I., Toint, P.L.: Trust Region Methods, vol. 1. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2000)

    Google Scholar 

  8. Fallahi, S., Salahi, M., Karbasy, S.A.: On SOCP/SDP formulation of the extended trust region subproblem (2018). arXiv:1807.07815

  9. Feng, J.M., Lin, G.X., Sheu, R.L., Xia, Y.: Duality and solutions for quadratic programming over single non-homogeneous quadratic constraint. J. Glob. Optim. 54(2), 275–293 (2012)

    Google Scholar 

  10. Hazan, E., Koren, T.: A linear-time algorithm for trust region problems. Math. Program. 1–19 (2015)

    Google Scholar 

  11. Ho-Nguyen, N., Kilinc-Karzan, F.: A second-order cone based approach for solving the trust-region subproblem and its variants. SIAM J. Optim. 27(3), 1485–1512 (2017)

    Google Scholar 

  12. Hsia, Y., Sheu, R.L.: Trust region subproblem with a fixed number of additional linear inequality constraints has polynomial complexity (2013). arXiv:1312.1398

  13. Jeyakumar, V., Li, G.: Trust-region problems with linear inequality constraints: exact SDP relaxation, global optimality and robust optimization. Math. Program. 147(1–2), 171–206 (2014)

    Google Scholar 

  14. Jiang, R., Li, D.: Novel reformulations and efficient algorithm for the generalized trust region subproblem (2017). arXiv:1707.08706

  15. Jiang, R., Li, D.: A linear-time algorithm for generalized trust region problems (2018). arXiv:1807.07563

  16. Jiang, R., Li, D.: Exactness conditions for SDP/SOCP relaxations of generalization of the extended trust region subproblem. Working paper (2019)

    Google Scholar 

  17. Jiang, R., Li, D., Wu, B.: SOCP reformulation for the generalized trust region subproblem via a canonical form of two symmetric matrices. Math. Program. 169(2), 531–563 (2018)

    Google Scholar 

  18. Locatelli, M.: Exactness conditions for an SDP relaxation of the extended trust region problem. Optim. Lett. 10(6), 1141–1151 (2016)

    Google Scholar 

  19. Moré, J.J.: Generalizations of the trust region problem. Optim. Methods Softw. 2(3–4), 189–209 (1993)

    Google Scholar 

  20. Moré, J.J., Sorensen, D.C.: Computing a trust region step. SIAM J. Sci. Stat. Comput. 4(3), 553–572 (1983)

    Google Scholar 

  21. Pardalos, P.M.: Global optimization algorithms for linearly constrained indefinite quadratic problems. Comput. Math. Appl. 21(6), 87–97 (1991)

    Google Scholar 

  22. Pólik, I., Terlaky, T.: A survey of the s-lemma. SIAM Rev. 49(3), 371–418 (2007)

    Google Scholar 

  23. Rendl, F., Wolkowicz, H.: A semidefinite framework for trust region subproblems with applications to large scale minimization. Math. Program. 77(1), 273–299 (1997)

    Google Scholar 

  24. Shor, N.Z.: Quadratic optimization problems. Sov. J. Comput. Syst. Sci. 25(6), 1–11 (1987)

    Google Scholar 

  25. Shor, N.: Dual quadratic estimates in polynomial and boolean programming. Ann. Oper. Res. 25(1), 163–168 (1990)

    Google Scholar 

  26. Stern, R.J., Wolkowicz, H.: Indefinite trust region subproblems and nonsymmetric eigenvalue perturbations. SIAM J. Optim. 5(2), 286–313 (1995)

    Google Scholar 

  27. Sturm, J.F., Zhang, S.: On cones of nonnegative quadratic functions. Math. Oper. Res. 28(2), 246–267 (2003)

    Google Scholar 

  28. Uhlig, F.: Definite and semidefinite matrices in a real symmetric matrix pencil. Pac. J. Math. 49(2), 561–568 (1973)

    Google Scholar 

  29. Yakubovich, V.A.: S-procedure in nonlinear control theory. Vestnik Leningrad University, vol. 1, pp. 62–77 (1971)

    Google Scholar 

  30. Ye, Y., Zhang, S.: New results on quadratic minimization. SIAM J. Optim. 14(1), 245–267 (2003)

    Google Scholar 

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Correspondence to Rujun Jiang .

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Jiang, R., Li, D. (2020). On Conic Relaxations of Generalization of the Extended Trust Region Subproblem. 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_15

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