SOCP reformulation for the generalized trust region subproblem via a canonical form of two symmetric matrices
We investigate in this paper the generalized trust region subproblem (GTRS) of minimizing a general quadratic objective function subject to a general quadratic inequality constraint. By applying a simultaneous block diagonalization approach, we obtain a congruent canonical form for the symmetric mat...
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| Published in: | Mathematical programming Vol. 169; no. 2; pp. 531 - 563 |
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| Abstract | We investigate in this paper the generalized trust region subproblem (GTRS) of minimizing a general quadratic objective function subject to a general quadratic inequality constraint. By applying a simultaneous block diagonalization approach, we obtain a congruent canonical form for the symmetric matrices in both the objective and constraint functions. By exploiting the block separability of the canonical form, we show that all GTRSs with an optimal value bounded from below are second order cone programming (SOCP) representable. Our result generalizes the recent work of Ben-Tal and den Hertog (Math. Program. 143(1–2):1–29,
2014
), which establishes the SOCP representability of the GTRS under the assumption of the simultaneous diagonalizability of the two matrices in the objective and constraint functions. We then derive a closed-form solution for the GTRS when the two matrices are not simultaneously diagonalizable. We further extend our method to two variants of the GTRS in which the inequality constraint is replaced by either an equality constraint or an interval constraint. |
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| AbstractList | We investigate in this paper the generalized trust region subproblem (GTRS) of minimizing a general quadratic objective function subject to a general quadratic inequality constraint. By applying a simultaneous block diagonalization approach, we obtain a congruent canonical form for the symmetric matrices in both the objective and constraint functions. By exploiting the block separability of the canonical form, we show that all GTRSs with an optimal value bounded from below are second order cone programming (SOCP) representable. Our result generalizes the recent work of Ben-Tal and den Hertog (Math. Program. 143(1–2):1–29, 2014), which establishes the SOCP representability of the GTRS under the assumption of the simultaneous diagonalizability of the two matrices in the objective and constraint functions. We then derive a closed-form solution for the GTRS when the two matrices are not simultaneously diagonalizable. We further extend our method to two variants of the GTRS in which the inequality constraint is replaced by either an equality constraint or an interval constraint. We investigate in this paper the generalized trust region subproblem (GTRS) of minimizing a general quadratic objective function subject to a general quadratic inequality constraint. By applying a simultaneous block diagonalization approach, we obtain a congruent canonical form for the symmetric matrices in both the objective and constraint functions. By exploiting the block separability of the canonical form, we show that all GTRSs with an optimal value bounded from below are second order cone programming (SOCP) representable. Our result generalizes the recent work of Ben-Tal and den Hertog (Math. Program. 143(1–2):1–29, 2014 ), which establishes the SOCP representability of the GTRS under the assumption of the simultaneous diagonalizability of the two matrices in the objective and constraint functions. We then derive a closed-form solution for the GTRS when the two matrices are not simultaneously diagonalizable. We further extend our method to two variants of the GTRS in which the inequality constraint is replaced by either an equality constraint or an interval constraint. |
| Author | Wu, Baiyi Jiang, Rujun Li, Duan |
| Author_xml | – sequence: 1 givenname: Rujun surname: Jiang fullname: Jiang, Rujun organization: Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong – sequence: 2 givenname: Duan surname: Li fullname: Li, Duan organization: Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong – sequence: 3 givenname: Baiyi surname: Wu fullname: Wu, Baiyi email: baiyiwu@outlook.com organization: School of Finance, Guangdong University of Foreign Studies |
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| Cites_doi | 10.1016/0024-3795(76)90066-5 10.1007/s10107-015-0907-0 10.1007/s10107-013-0710-8 10.1145/355900.355912 10.1007/BF02614438 10.1137/S105262340139001X 10.1137/1.9780898717655 10.1137/1.9780898718829 10.1007/s10589-013-9635-7 10.1137/1.9780898719857 10.1007/BF01580852 10.1137/0804009 10.1287/moor.28.2.246.14485 10.1007/s10898-010-9625-6 10.1007/BF02592331 10.1137/S003614450444556X 10.1080/10556789308805542 10.1137/15M1023920 10.1137/0805016 10.1007/s11590-014-0812-0 10.1007/s40314-016-0349-1 10.1137/0904038 10.1137/S003614450444614X |
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| Keywords | 90C20 90C26 Trust region subproblem Quadratically constrained quadratic programming Canonical form |
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| SubjectTerms | Calculus of Variations and Optimal Control; Optimization Combinatorics Economic models Full Length Paper Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Matrix methods Numerical Analysis Theoretical |
| Title | SOCP reformulation for the generalized trust region subproblem via a canonical form of two symmetric matrices |
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