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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Vydáno v:	Mathematical programming Ročník 169; číslo 2; s. 531 - 563
Hlavní autoři:	Jiang, Rujun, Li, Duan, Wu, Baiyi
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2018 Springer Nature B.V
Témata:	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 90C20 90C26 Trust region subproblem Quadratically constrained quadratic programming Canonical form
ISSN:	0025-5610, 1436-4646
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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.
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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Copyright	Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2017 Mathematical Programming is a copyright of Springer, (2017). All Rights Reserved.
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References	SternRJWolkowiczHIndefinite trust region subproblems and nonsymmetric eigenvalue perturbationsSIAM J. Optim.199552286313133019910.1137/08050160846.49017 Adachi, S., Nakatsukasa, Y.: Eigenvalue-based algorithm and analysis for nonconvex QCQP with one constraint (2016). http://www.keisu.t.u-tokyo.ac.jp/research/techrep/data/2016/METR16-07.pdf LancasterPRodmanLCanonical forms for hermitian matrix pairs under strict equivalence and congruenceSIAM Rev.2005473407443217863510.1137/S003614450444556X1087.15014 MoréJJSorensenDCComputing a trust region stepSIAM J. Sci. Stat. Comput.19834355357272311010.1137/09040380551.65042 RendlFWolkowiczHA semidefinite framework for trust region subproblems with applications to large scale minimizationMath. Program1997771273299146138410.1007/BF026144380888.90137 MoréJJGeneralizations of the trust region problemOptim. Methods Softw.199323–418920910.1080/10556789308805542 HsiaYLinGXSheuRLA revisit to quadratic programming with one inequality quadratic constraint via matrix pencilPac. J. Optim.201410346148132486811327.90168 GolubGHVan LoanCFMatrix Computations2012BaltimoreJHU Press1268.65037 PólikITerlakyTA survey of the S-lemmaSIAM Rev.2007493371418235380410.1137/S003614450444614X1128.90046 SturmJFZhangSOn cones of nonnegative quadratic functionsMath. Oper. Res.2003282246267198066210.1287/moor.28.2.246.144851082.90086 XiaYWangSSheuRLS-lemma with equality and its applicationsMath. Program20161561–2513547345920910.1007/s10107-015-0907-01333.90086 YuanYOn a subproblem of trust region algorithms for constrained optimizationMath. Program1990471–35363105484110.1007/BF015808520711.90062 MartínezJMLocal minimizers of quadratic functions on Euclidean balls and spheresSIAM J. Optim.199441159176126041310.1137/08040090801.65057 WangSXiaYStrong duality for generalized trust region subproblem: S-lemma with interval boundsOptim. Lett.20159610631073337366810.1007/s11590-014-0812-01354.90089 Salahi, M., Taati, A.: An efficient algorithm for solving the generalized trust region subproblem. Comp. Appl. Math. (2016). doi:10.1007/s40314-016-0349-1 JiangRLiDSimultaneous diagonalization of matrices and its applications in quadratically constrained quadratic programmingSIAM J. Optim.201626316491668353789110.1137/15M10239201347.65107 PongTKWolkowiczHThe generalized trust region subproblemComput. Optim. Appl.2014582273322320196310.1007/s10589-013-9635-71329.90100 Ben-TalAden HertogDHidden conic quadratic representation of some nonconvex quadratic optimization problemsMath. Program20141431–2129315206110.1007/s10107-013-0710-81295.90036 GuoCHHighamNJTisseurFAn improved arc algorithm for detecting definite hermitian pairsSIAM J. Matrix Anal. Appl.200931311311151255881610.1137/08074218X1202.65054 Ben-TalANemirovskiALectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications2001PhiladelphiaSIAM10.1137/1.97808987188290986.90032 ConnARGouldNITointPLTrust Region Methods2000PhiladelphiaSIAM10.1137/1.97808987198570958.65071 Hmam, H.: Quadratic optimization with one quadratic equality constraint. Tech. Rep., Warfare and Radar Division DSTO Defence Science and Technology Organisation, Report DSTO-TR-2416 (2010) FengJMLinGXSheuRLXiaYDuality and solutions for quadratic programming over single non-homogeneous quadratic constraintJ. Global Optim.2012542275293297962910.1007/s10898-010-9625-61281.90032 UhligFA canonical form for a pair of real symmetric matrices that generate a nonsingular pencilLinear Algebra Appl.197614318920957301110.1016/0024-3795(76)90066-50338.15009 Ben-TalATeboulleMHidden convexity in some nonconvex quadratically constrained quadratic programmingMath. Program19967215163138516310.1007/BF025923310851.90087 KågströmBRuheAAn algorithm for numerical computation of the Jordan normal form of a complex matrixACM Trans. Math. Softw.19806339841958534710.1145/355900.3559120434.65020 YeYZhangSNew results on quadratic minimizationSIAM J. Optim.2003141245267200594310.1137/S105262340139001X1043.90064 DattaBNNumerical Linear Algebra and Applications2010PhiladelphiaSIAM10.1137/1.97808987176551187.65027 CH Guo (1145_CR9) 2009; 31 JM Feng (1145_CR7) 2012; 54 B Kågström (1145_CR13) 1980; 6 RJ Stern (1145_CR22) 1995; 5 GH Golub (1145_CR8) 2012 Y Xia (1145_CR26) 2016; 156 Y Yuan (1145_CR28) 1990; 47 JF Sturm (1145_CR23) 2003; 28 1145_CR21 JJ Moré (1145_CR16) 1993; 2 S Wang (1145_CR25) 2015; 9 F Rendl (1145_CR20) 1997; 77 BN Datta (1145_CR6) 2010 A Ben-Tal (1145_CR3) 2001 Y Hsia (1145_CR11) 2014; 10 I Pólik (1145_CR18) 2007; 49 Y Ye (1145_CR27) 2003; 14 JJ Moré (1145_CR17) 1983; 4 F Uhlig (1145_CR24) 1976; 14 A Ben-Tal (1145_CR4) 1996; 72 R Jiang (1145_CR12) 2016; 26 JM Martínez (1145_CR15) 1994; 4 1145_CR10 AR Conn (1145_CR5) 2000 1145_CR1 P Lancaster (1145_CR14) 2005; 47 A Ben-Tal (1145_CR2) 2014; 143 TK Pong (1145_CR19) 2014; 58
References_xml	– reference: Hmam, H.: Quadratic optimization with one quadratic equality constraint. Tech. Rep., Warfare and Radar Division DSTO Defence Science and Technology Organisation, Report DSTO-TR-2416 (2010) – reference: HsiaYLinGXSheuRLA revisit to quadratic programming with one inequality quadratic constraint via matrix pencilPac. J. Optim.201410346148132486811327.90168 – reference: DattaBNNumerical Linear Algebra and Applications2010PhiladelphiaSIAM10.1137/1.97808987176551187.65027 – reference: JiangRLiDSimultaneous diagonalization of matrices and its applications in quadratically constrained quadratic programmingSIAM J. Optim.201626316491668353789110.1137/15M10239201347.65107 – reference: YuanYOn a subproblem of trust region algorithms for constrained optimizationMath. Program1990471–35363105484110.1007/BF015808520711.90062 – reference: GolubGHVan LoanCFMatrix Computations2012BaltimoreJHU Press1268.65037 – reference: KågströmBRuheAAn algorithm for numerical computation of the Jordan normal form of a complex matrixACM Trans. Math. Softw.19806339841958534710.1145/355900.3559120434.65020 – reference: MartínezJMLocal minimizers of quadratic functions on Euclidean balls and spheresSIAM J. Optim.199441159176126041310.1137/08040090801.65057 – reference: PólikITerlakyTA survey of the S-lemmaSIAM Rev.2007493371418235380410.1137/S003614450444614X1128.90046 – reference: ConnARGouldNITointPLTrust Region Methods2000PhiladelphiaSIAM10.1137/1.97808987198570958.65071 – reference: FengJMLinGXSheuRLXiaYDuality and solutions for quadratic programming over single non-homogeneous quadratic constraintJ. Global Optim.2012542275293297962910.1007/s10898-010-9625-61281.90032 – reference: Salahi, M., Taati, A.: An efficient algorithm for solving the generalized trust region subproblem. Comp. Appl. Math. (2016). doi:10.1007/s40314-016-0349-1 – reference: Ben-TalAden HertogDHidden conic quadratic representation of some nonconvex quadratic optimization problemsMath. Program20141431–2129315206110.1007/s10107-013-0710-81295.90036 – reference: SturmJFZhangSOn cones of nonnegative quadratic functionsMath. Oper. Res.2003282246267198066210.1287/moor.28.2.246.144851082.90086 – reference: XiaYWangSSheuRLS-lemma with equality and its applicationsMath. Program20161561–2513547345920910.1007/s10107-015-0907-01333.90086 – reference: MoréJJGeneralizations of the trust region problemOptim. Methods Softw.199323–418920910.1080/10556789308805542 – reference: YeYZhangSNew results on quadratic minimizationSIAM J. 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Snippet	We investigate in this paper the generalized trust region subproblem (GTRS) of minimizing a general quadratic objective function subject to a general quadratic...
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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
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Title	SOCP reformulation for the generalized trust region subproblem via a canonical form of two symmetric matrices
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