메뉴 건너뛰기




Volumn 129, Issue 1, 2011, Pages 129-157

Semidefinite relaxations for quadratically constrained quadratic programming: A review and comparisons

Author keywords

Lagrangian relaxation; Nonconvex optimization; Quadratic programming; Semidefinite programming

Indexed keywords

COMPUTATIONAL COMPARISONS; COMPUTATIONAL TIME; CONSTRAINED QUADRATIC PROGRAMMING; FIRST-ORDER; LAGRANGIAN RELAXATION; NON-LINEAR CONSTRAINTS; NONCONVEX OPTIMIZATION; SDP RELAXATION; SEMI-DEFINITE PROGRAMMING; SEMIDEFINITE PROGRAMS; SEMIDEFINITE RELAXATION; THREE ORDERS OF MAGNITUDE;

EID: 80052586194     PISSN: 00255610     EISSN: 14364646     Source Type: Journal    
DOI: 10.1007/s10107-011-0462-2     Document Type: Article
Times cited : (140)

References (50)
  • 2
    • 0026904595 scopus 로고
    • Generalized bilinear programming: Part I. Models, applications and linear programming relaxation
    • 0784.90051 10.1016/0377-2217(92)90082-K
    • F.A. Al-Khayyal 1992 Generalized bilinear programming: Part I. Models, applications and linear programming relaxation Eur. J. Oper. Res. 60 306 314 0784.90051 10.1016/0377-2217(92)90082-K
    • (1992) Eur. J. Oper. Res. , vol.60 , pp. 306-314
    • Al-Khayyal, F.A.1
  • 3
    • 0020752471 scopus 로고
    • Jointly constrained biconvex programming
    • 707057 0521.90087 10.1287/moor.8.2.273
    • F.A. Al-Khayyal J.E. Falk 1983 Jointly constrained biconvex programming Math. Oper. Res. 8 273 286 707057 0521.90087 10.1287/moor.8.2.273
    • (1983) Math. Oper. Res. , vol.8 , pp. 273-286
    • Al-Khayyal, F.A.1    Falk, J.E.2
  • 4
    • 0000483159 scopus 로고
    • A relaxation method for nonconvex quadratically constrained quadratic programs
    • 1327189 0835.90060 10.1007/BF01099462
    • F.A. Al-Khayyal C. Larsen T. Van Voorhis 1995 A relaxation method for nonconvex quadratically constrained quadratic programs J. Global Optim. 6 215 230 1327189 0835.90060 10.1007/BF01099462
    • (1995) J. Global Optim. , vol.6 , pp. 215-230
    • Al-Khayyal, F.A.1    Larsen, C.2    Van Voorhis, T.3
  • 6
    • 59449105344 scopus 로고    scopus 로고
    • Semidefinite programming versus the reformulation-linearization technique for nonconvex quadratically constrained quadratic programming
    • 2471898 1169.90425 10.1007/s10898-008-9372-0
    • K.M. Anstreicher 2009 Semidefinite programming versus the reformulation-linearization technique for nonconvex quadratically constrained quadratic programming J. Global Optim. 43 471 484 2471898 1169.90425 10.1007/s10898-008-9372-0
    • (2009) J. Global Optim. , vol.43 , pp. 471-484
    • Anstreicher, K.M.1
  • 7
    • 27744568236 scopus 로고    scopus 로고
    • D.C. versus copositive bounds for standard QP
    • DOI 10.1007/s10898-004-4312-0
    • K.M. Anstreicher S. Burer 2005 DC versus copositive bounds for standard QP J. Global Optim. 33 2 299 312 2192210 1093.90033 10.1007/s10898-004-4312-0 (Pubitemid 41618777)
    • (2005) Journal of Global Optimization , vol.33 , Issue.2 , pp. 299-312
    • Anstreicher, K.M.1    Burer, S.2
  • 8
    • 68949144575 scopus 로고    scopus 로고
    • Multiterm polyhedral relaxations for nonconvex, quadratically-constrained quadratic programs
    • 2554897 1179.90252 10.1080/10556780902883184
    • X. Bao N.V. Sahinidis M. Tawarmalani 2009 Multiterm polyhedral relaxations for nonconvex, quadratically-constrained quadratic programs Optim. Methods Softw. 24 485 504 2554897 1179.90252 10.1080/10556780902883184
    • (2009) Optim. Methods Softw. , vol.24 , pp. 485-504
    • Bao, X.1    Sahinidis, N.V.2    Tawarmalani, M.3
  • 10
    • 39349095966 scopus 로고    scopus 로고
    • DSDP5: Software for semidefinite programming
    • Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL. ACM Trans. Math. Softw.
    • Benson, S.J., Ye, Y.: DSDP5: Software for semidefinite programming. Tech. Rep. ANL/MCS-P1289-0905, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL. ACM Trans. Math. Softw. http://www.mcs.anl. gov/~benson/dsdp (submitted, 2005)
    • Tech. Rep. ANL/MCS-P1289-0905
    • Benson S., .J.1    Ye, Y.2
  • 11
    • 31244435061 scopus 로고    scopus 로고
    • Solving standard quadratic optimization problems via linear, semidefinite and copositive programming
    • DOI 10.1023/A:1020209017701
    • I.M. Bomze E. de Klerk 2002 Solving standard quadratic optimization problems via linear, semidefinite and copositive programming J. Global Optim. 24 163 185 1934026 1047.90038 10.1023/A:1020209017701 (Pubitemid 35168547)
    • (2002) Journal of Global Optimization , vol.24 , Issue.2
    • Bomze, I.M.1    De Klerk, E.2
  • 12
    • 0011565238 scopus 로고    scopus 로고
    • On copositive programming and standard quadratic optimization problems
    • 1810401 0970.90057 10.1023/A:1026583532263
    • I.M. Bomze M. Dur E. de Klerk C. Roos A.J. Quist T. Terlaky 2000 On copositive programming and standard quadratic optimization problems J. Global Optim. 18 4 301 320 1810401 0970.90057 10.1023/A:1026583532263
    • (2000) J. Global Optim. , vol.18 , Issue.4 , pp. 301-320
    • Bomze, I.M.1    Dur, M.2    De Klerk, E.3    Roos, C.4    Quist, A.J.5    Terlaky, T.6
  • 13
    • 49749147099 scopus 로고    scopus 로고
    • Improved SDP bounds for minimizing quadratic functions over the l(1)-ball
    • 2357607 1135.90377 10.1007/s11590-006-0018-1
    • I.M. Bomze F. Frommlet M. Rubey 2007 Improved SDP bounds for minimizing quadratic functions over the l(1)-ball Optim. Lett. 1 1 49 59 2357607 1135.90377 10.1007/s11590-006-0018-1
    • (2007) Optim. Lett. , vol.1 , Issue.1 , pp. 49-59
    • Bomze, I.M.1    Frommlet, F.2    Rubey, M.3
  • 14
    • 43949093438 scopus 로고    scopus 로고
    • New and old bounds for standard quadratic optimization: Dominance, equivalence and incomparability
    • DOI 10.1007/s10107-007-0138-0
    • I.M. Bomze M. Locatelli F. Tardella 2008 New and old bounds for standard quadratic optimization: dominance, equivalence and incomparability Math. Program. 115 1 31 64 2403752 1171.90007 10.1007/s10107-007-0138-0 (Pubitemid 351702151)
    • (2008) Mathematical Programming , vol.115 , Issue.1 , pp. 31-64
    • Bomze, I.M.1    Locatelli, M.2    Tardella, F.3
  • 15
    • 0033295981 scopus 로고    scopus 로고
    • CSDP, A C library for semidefinite programming
    • 1778432 10.1080/10556789908805765
    • B. Borchers 1999 CSDP, A C library for semidefinite programming Optim. Methods Softw. 11 613 623 1778432 10.1080/10556789908805765
    • (1999) Optim. Methods Softw. , vol.11 , pp. 613-623
    • Borchers, B.1
  • 17
    • 80052585863 scopus 로고    scopus 로고
    • Optimizing a polyhedral-semidefinite relaxation of completely positive programs
    • University of Iowa, Iowa City, IA
    • Burer, S.: Optimizing a polyhedral-semidefinite relaxation of completely positive programs. Tech. rep., University of Iowa, Iowa City, IA. http://www.optimization-online.org/DB-FILE/2009/01/2184.pdf (2008)
    • Tech. Rep.
    • Burer, S.1
  • 18
    • 67349282710 scopus 로고    scopus 로고
    • On the copositive representation of binary and continuous nonconvex quadratic programs
    • 2505747 10.1007/s10107-008-0223-z
    • S. Burer 2009 On the copositive representation of binary and continuous nonconvex quadratic programs Math. Program. 120 2 295 479 2505747 10.1007/s10107-008-0223-z
    • (2009) Math. Program. , vol.120 , Issue.2 , pp. 295-479
    • Burer, S.1
  • 19
    • 80052574798 scopus 로고    scopus 로고
    • Relaxing the optimality conditions of box QP
    • Dept. of Management Sciences, University of Iowa, Iowa City, IA 52240
    • Burer, S., Chen, J.: Relaxing the optimality conditions of box QP. Tech. rep., Dept. of Management Sciences, University of Iowa, Iowa City, IA 52240. Available at http://www.optimization-online.org/DB-FILE/2007/10/1815.pdf (2007)
    • Tech. Rep.
    • Burer, S.1    Chen, J.2
  • 20
    • 38749116247 scopus 로고    scopus 로고
    • A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations
    • DOI 10.1007/s10107-006-0080-6
    • S. Burer D. Vandenbussche 2008 A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations Math. Program. 112 259 282 2375483 10.1007/s10107-006-0080-6 (Pubitemid 351177972)
    • (2008) Mathematical Programming , vol.113 , Issue.2 , pp. 259-282
    • Burer, S.1    Vandenbussche, D.2
  • 21
    • 0038292295 scopus 로고
    • Global optimization algorithms for chip layout and compaction
    • 10.1080/03052159508941259
    • M.C. Dorneich N.V. Sahinidis 1995 Global optimization algorithms for chip layout and compaction Eng. Optim. 25 131 154 10.1080/03052159508941259
    • (1995) Eng. Optim. , vol.25 , pp. 131-154
    • Dorneich, M.C.1    Sahinidis, N.V.2
  • 22
    • 0000201892 scopus 로고    scopus 로고
    • Semidefinite Programming Relaxation for Nonconvex Quadratic Programs
    • T. Fujie M. Kojima 1997 Semidefinite programming relaxation for nonconvex quadratic programs J. Global Optim. 10 367 380 1457183 0881.90101 10.1023/A:1008282830093 (Pubitemid 127502196)
    • (1997) Journal of Global Optimization , vol.10 , Issue.4 , pp. 367-380
    • Fujie, T.1    Kojima, M.2
  • 23
    • 84893574327 scopus 로고
    • Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming
    • 1412228 0885.68088 10.1145/227683.227684
    • M.X. Goemans D.P. Williamson 1995 Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming J. ACM 42 1115 1145 1412228 0885.68088 10.1145/227683.227684
    • (1995) J. ACM , vol.42 , pp. 1115-1145
    • Goemans, M.X.1    Williamson, D.P.2
  • 24
    • 0039361444 scopus 로고
    • Quadratically constrained quadratic programming: Some applications and a method for solution
    • 0479.90065
    • E.P. Hao 1982 Quadratically constrained quadratic programming: some applications and a method for solution Math. Methods Oper. Res. 26 105 119 0479.90065
    • (1982) Math. Methods Oper. Res. , vol.26 , pp. 105-119
    • Hao, E.P.1
  • 25
    • 21144432296 scopus 로고    scopus 로고
    • A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs
    • DOI 10.1007/s10107-005-0582-7
    • J. Linderoth 2005 A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs Math. Program. 103 251 282 2146380 1099.90039 10.1007/s10107-005-0582-7 (Pubitemid 40878750)
    • (2005) Mathematical Programming , vol.103 , Issue.2 , pp. 251-282
    • Linderoth, J.1
  • 26
    • 0031187688 scopus 로고    scopus 로고
    • Process planning in a fuzzy environment
    • PII S0377221796000252
    • M.L. Liu N.V. Sahinidis 1997 Process planning in a fuzzy environment Eur. J. Oper. Res. 100 142 169 0947.90608 10.1016/S0377-2217(96)00025-2 (Pubitemid 127414916)
    • (1997) European Journal of Operational Research , vol.100 , Issue.1 , pp. 142-169
    • Liu, M.L.1    Sahinidis, N.V.2
  • 27
    • 84867958818 scopus 로고    scopus 로고
    • Packing equal circles in a square: A deterministic global optimization approach
    • DOI 10.1016/S0166-218X(01)00359-6, PII S0166218X01003596
    • M. Locatelli U. Raber 2002 Packing equal circles in a square: a deterministic global optimization approach Discret. Appl. Math. 122 139 166 1907828 1019.90033 10.1016/S0166-218X(01)00359-6 (Pubitemid 34548371)
    • (2002) Discrete Applied Mathematics , vol.122 , Issue.1-3
    • Locatelli, M.1    Raber, U.2
  • 28
    • 34250381644 scopus 로고
    • Computability of global solutions to factorable nonconvex programs: Part I-Convex underestimating problems
    • 469281 0349.90100 10.1007/BF01580665
    • G.P. McCormick 1976 Computability of global solutions to factorable nonconvex programs: Part I-Convex underestimating problems Math. Program. 10 147 175 469281 0349.90100 10.1007/BF01580665
    • (1976) Math. Program. , vol.10 , pp. 147-175
    • McCormick, G.P.1
  • 29
    • 3142688043 scopus 로고    scopus 로고
    • An independent benchmarking of SDP and SOCP solvers
    • DOI 10.1007/s10107-002-0355-5
    • H.D. Mittelmann 2003 An independent benchmarking of SDP and SOCP solvers Math. Program. 95 407 430 1976487 1030.90080 10.1007/s10107-002-0355-5 (Pubitemid 44757861)
    • (2003) Mathematical Programming, Series B , vol.95 , Issue.2 , pp. 407-430
    • Mittelmann, H.D.1
  • 30
    • 0002376103 scopus 로고    scopus 로고
    • A New Semidefinite Programming Bound for Indefinite Quadratic Forms over a Simplex
    • I. Nowak 1999 A new semidefinite programming bound for indefinite quadratic forms over a simplex J. Global Optim. 14 357 364 1707796 0959.65077 10.1023/A:1008315627883 (Pubitemid 129617301)
    • (1999) Journal of Global Optimization , vol.14 , Issue.4 , pp. 357-364
    • Nowak, I.1
  • 31
    • 34548491396 scopus 로고    scopus 로고
    • A survey of the S-lemma
    • DOI 10.1137/S003614450444614X
    • I. Pólik T. Terlaky 2007 A survey of the S-Lemma SIAM Rev. 49 371 418 2353804 1128.90046 10.1137/S003614450444614X (Pubitemid 47377690)
    • (2007) SIAM Review , vol.49 , Issue.3 , pp. 371-418
    • Polik, I.1    Terlaky, T.2
  • 32
    • 0002821681 scopus 로고
    • A recipe for semidefinite relaxation for (0,1)-quadratic programming
    • 1342935 0843.90088 10.1007/BF01100205
    • S. Poljak F. Rendl H. Wolkowicz 1995 A recipe for semidefinite relaxation for (0,1)-quadratic programming J. Global Optim. 7 1 51 73 1342935 0843.90088 10.1007/BF01100205
    • (1995) J. Global Optim. , vol.7 , Issue.1 , pp. 51-73
    • Poljak, S.1    Rendl, F.2    Wolkowicz, H.3
  • 33
    • 0031674391 scopus 로고    scopus 로고
    • Copositive relaxation for general quadratic programming
    • A.J. Quist E. de Klerk C. Roos T. Terlaky 1998 Copositive relaxations for general quadratic programming Optim. Methods Softw. 9 185 208 1618092 0904.90126 10.1080/10556789808805692 (Pubitemid 128542196)
    • (1998) Optimization Methods and Software , vol.9 , Issue.1-3 , pp. 185-208
    • Quist, A.J.1    De Klerk, E.2    Roos, C.3    Terlaky, T.4
  • 34
    • 0013490621 scopus 로고    scopus 로고
    • A Convex Envelope Formula for Multilinear Functions
    • A.D. Rikun 1997 A convex envelope formula for multilinear functions J. Global Optim. 10 425 437 1457187 0881.90099 10.1023/A:1008217604285 (Pubitemid 127502200)
    • (1997) Journal of Global Optimization , vol.10 , Issue.4 , pp. 425-437
    • Rikun, A.D.1
  • 36
    • 0346226520 scopus 로고    scopus 로고
    • Convex envelopes of multilinear functions over a unit hypercube and over special discrete sets
    • 1479749 0914.90205
    • H.D. Sherali 1997 Convex envelopes of multilinear functions over a unit hypercube and over special discrete sets Acta Mathematica Vietnamica 22 245 270 1479749 0914.90205
    • (1997) Acta Mathematica Vietnamica , vol.22 , pp. 245-270
    • Sherali, H.D.1
  • 37
    • 33846874152 scopus 로고    scopus 로고
    • RLT: A unified approach for discrete and continuous nonconvex optimization
    • 2313377 1213.90029 10.1007/s10479-006-0107-7
    • H.D. Sherali 2007 RLT: a unified approach for discrete and continuous nonconvex optimization Ann. Oper. Res. 149 1 185 193 2313377 1213.90029 10.1007/s10479-006-0107-7
    • (2007) Ann. Oper. Res. , vol.149 , Issue.1 , pp. 185-193
    • Sherali, H.D.1
  • 38
    • 0023454146 scopus 로고
    • Quadratic optimization problems
    • 939596 0655.90055
    • N.Z. Shor 1987 Quadratic optimization problems Soviet J. Comput. Syst. Sci. 25 1 11 939596 0655.90055
    • (1987) Soviet J. Comput. Syst. Sci. , vol.25 , pp. 1-11
    • Shor, N.Z.1
  • 39
    • 0000834224 scopus 로고
    • Dual quadratic estimates in polynomial and Boolean programming
    • 1084428 0783.90081 10.1007/BF02283692
    • N.Z. Shor 1990 Dual quadratic estimates in polynomial and Boolean programming Ann. Oper. Res. 25 163 168 1084428 0783.90081 10.1007/BF02283692
    • (1990) Ann. Oper. Res. , vol.25 , pp. 163-168
    • Shor, N.Z.1
  • 40
    • 0002909079 scopus 로고
    • Dual estimates in multiextremal problems
    • 1263608 0765.90072 10.1007/BF00122430
    • N.Z. Shor 1992 Dual estimates in multiextremal problems J. Global Optim. 2 411 418 1263608 0765.90072 10.1007/BF00122430
    • (1992) J. Global Optim. , vol.2 , pp. 411-418
    • Shor, N.Z.1
  • 41
    • 9744246875 scopus 로고    scopus 로고
    • Global optimization approach to unequal global optimization approach to unequal sphere packing problems in 3D
    • DOI 10.1023/A:1016083231326
    • A. Sutou Y. Dai 2002 Global optimization approach to unequal sphere packing problems in 3D J. Optim. Theory Appl. 114 671 694 1921172 1026.90078 10.1023/A:1016083231326 (Pubitemid 39587619)
    • (2002) Journal of Optimization Theory and Applications , vol.114 , Issue.3 , pp. 671-694
    • Sutou, A.1    Dai, Y.2
  • 42
    • 21144437792 scopus 로고    scopus 로고
    • A polyhedral branch-and-cut approach to global optimization
    • DOI 10.1007/s10107-005-0581-8
    • M. Tawarmalani N.V. Sahinidis 2005 A polyhedral branch-and-cut approach to global optimization Math. Program. 103 225 249 2146379 1099.90047 10.1007/s10107-005-0581-8 (Pubitemid 40878749)
    • (2005) Mathematical Programming , vol.103 , Issue.2 , pp. 225-249
    • Tawarmalani, M.1    Sahinidis, N.V.2
  • 43
    • 31244432959 scopus 로고    scopus 로고
    • A global optimization algorithm using Lagrangian underestimates and the interval Newton method
    • DOI 10.1023/A:1020383700229
    • T. Van Voorhis 2002 A global optimization algorithm using lagrangian underestimates and the interval Newton method J. Global Optim. 24 349 370 1932387 1046.90067 10.1023/A:1020383700229 (Pubitemid 35254533)
    • (2002) Journal of Global Optimization , vol.24 , Issue.3
    • Van Voorhis, T.1
  • 44
    • 0025505751 scopus 로고
    • Quadratic programming is in NP
    • 1079809 0719.90052 10.1016/0020-0190(90)90100-C
    • S.A. Vavasis 1990 Quadratic programming is in NP Inf. Process. Lett. 36 73 77 1079809 0719.90052 10.1016/0020-0190(90)90100-C
    • (1990) Inf. Process. Lett. , vol.36 , pp. 73-77
    • Vavasis, S.A.1
  • 45
    • 0005244597 scopus 로고    scopus 로고
    • Semidefinite and lagrangian relaxations for hard combinatorial problems
    • Kluwer, B.V., Deventer, The Netherlands
    • Wolkowicz, H.: Semidefinite and lagrangian relaxations for hard combinatorial problems. In: Proceedings of the 19th IFIP TC7 Conference on System Modelling and Optimization, pp. 269-310. Kluwer, B.V., Deventer, The Netherlands (2000)
    • (2000) Proceedings of the 19th IFIP TC7 Conference on System Modelling and Optimization , pp. 269-310
    • Wolkowicz, H.1
  • 46
    • 0036883766 scopus 로고    scopus 로고
    • A note on lack of strong duality for quadratic problems with orthogonal constraints
    • DOI 10.1016/S0377-2217(02)00295-3, PII S0377221702002953
    • H. Wolkowicz 2002 A note on lack of strong duality for quadratic problems with orthogonal constraints Eur. J. Oper. Res. 143 2 356 364 1940977 1058.90046 10.1016/S0377-2217(02)00295-3 (Pubitemid 34991906)
    • (2002) European Journal of Operational Research , vol.143 , Issue.2 , pp. 356-364
    • Wolkowicz, H.1
  • 47
    • 84867957035 scopus 로고    scopus 로고
    • Semidefinite programming for discrete optimization and matrix completion problems
    • DOI 10.1016/S0166-218X(01)00352-3, PII S0166218X01003523
    • H. Wolkowicz M.F. Anjos 2002 Semidefinite programming for discrete optimization and matrix completion problems Discret. Appl. Math. 123 1-3 513 577 1922345 1060.90059 10.1016/S0166-218X(01)00352-3 (Pubitemid 34775584)
    • (2002) Discrete Applied Mathematics , vol.123 , Issue.1-3
    • Wolkowicz, H.1    Anjos, M.2
  • 49
    • 38849134090 scopus 로고    scopus 로고
    • A branch-and-bound algorithm for the continuous facility layout problem
    • DOI 10.1016/j.compchemeng.2007.05.003, PII S0098135407001354
    • W. Xie N.V. Sahinidis 2008 A branch-and-bound algorithm for the continuous facility layout problem Comput. Chem. Eng. 32 1016 1028 10.1016/j.compchemeng.2007.05.003 (Pubitemid 351199688)
    • (2008) Computers and Chemical Engineering , vol.32 , Issue.4-5 , pp. 1016-1028
    • Xie, W.1    Sahinidis, N.V.2
  • 50
    • 0042486152 scopus 로고    scopus 로고
    • Implementation and evaluation of SDPA 6.0 (SemiDefinite Programming Algorithm 6.0)
    • 2019042 1106.90366 10.1080/1055678031000118482
    • M. Yamashita K. Fujisawa M. Kojima 2003 Implementation and evaluation of SDPA 6.0 (SemiDefinite Programming Algorithm 6.0) Optim. Methods Softw. 18 491 505 2019042 1106.90366 10.1080/1055678031000118482
    • (2003) Optim. Methods Softw. , vol.18 , pp. 491-505
    • Yamashita, M.1    Fujisawa, K.2    Kojima, M.3


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.