메뉴 건너뛰기




Volumn 68, Issue 1, 2011, Pages 53-102

Euclidean distance matrices, semidefinite programming and sensor network localization

Author keywords

Euclidean distance matrix completions; Fundamental problem of distance geometry; Semidefinite programming; Sensor network localization

Indexed keywords


EID: 84862705457     PISSN: 00325155     EISSN: 16622758     Source Type: Journal    
DOI: 10.4171/PM/1881     Document Type: Article
Times cited : (24)

References (109)
  • 2
    • 0002740666 scopus 로고    scopus 로고
    • Solving Euclidean distance matrix completion problems via semidefinite programming
    • A. Y. Alfakih, A. Khandani, and H. Wolkowicz, Solving Euclidean distance matrix completion problems via semidefinite programming. Comput. Optim. Appl. 12 (1999), 13-30. Zbl 1040.90537 MR 1704098
    • (1999) Comput. Optim. Appl , vol.12 , pp. 13-30
    • Alfakih, A.Y.1    Khandani, A.2    Wolkowicz, H.3
  • 5
    • 23044487729 scopus 로고    scopus 로고
    • Approximate and exact completion problems for Euclidean distance matrices using semidefinite programming
    • S. Al-Homidan and H. Wolkowicz, Approximate and exact completion problems for Euclidean distance matrices using semidefinite programming. Linear Algebra Appl. 406 (2005), 109-141. Zbl 1081.15011 MR 2156431
    • (2005) Linear Algebra Appl , vol.406 , pp. 109-141
    • Al-Homidan, S.1    Wolkowicz, H.2
  • 6
    • 33645443112 scopus 로고
    • The rigidity of graphs
    • L. Asimow and B. Roth, The rigidity of graphs. Trans. Amer. Math. Soc. 245 (1978), 279-289. Zbl 0392.05026 MR 511410
    • (1978) Trans. Amer. Math. Soc , vol.245 , pp. 279-289
    • Asimow, L.1    Roth, B.2
  • 7
    • 0018444223 scopus 로고
    • The rigidity of graphs
    • L. Asimow and B. Roth, The rigidity of graphs. II. J. Math. Anal. Appl. 68 (1979), 171-190. Zbl 0441.05046 MR 531431
    • (1979) II. J. Math. Anal. Appl , vol.68 , pp. 171-190
    • Asimow, L.1    Roth, B.2
  • 9
    • 21844519147 scopus 로고
    • The Euclidean distance matrix completion problem
    • M. Bakonyi and C. R. Johnson, The Euclidean distance matrix completion problem. SIAM J. Matrix Anal. Appl. 16 (1995), 646-654. Zbl 0823.15012 MR 1321802
    • (1995) SIAM J. Matrix Anal. Appl , vol.16 , pp. 646-654
    • Bakonyi, M.1    Johnson, C.R.2
  • 10
    • 33746874045 scopus 로고    scopus 로고
    • Semidefinite programming based algorithms for sensor network localization
    • P. Biswas, T.-C. Lian, T.-C. Wang, and Y. Ye. Semidefinite programming based algorithms for sensor network localization. ACM Trans. Sensor Network 2 (2006), 188-220.
    • (2006) ACM Trans. Sensor Network , vol.2 , pp. 188-220
    • Biswas, P.1    Lian, T.-C.2    Wang, T.-C.3    Ye, Y.4
  • 11
    • 33750083077 scopus 로고    scopus 로고
    • Semidefinite programming approaches for sensor network localization with noisy distance measurements IEEE Trans
    • P. Biswas, T.-C. Liang, K.-C. Toh, Y. Ye, and T.-C. Wang, Semidefinite program- ming approaches for sensor network localization with noisy distance measurements IEEE Trans. Autom. Sci. Eng. 3 (2006), 360-371.
    • (2006) Autom. Sci. Eng , vol.3 , pp. 360-371
    • Biswas, P.1    Liang, T.-C.2    Toh, K.-C.3    Ye, Y.4    Wang, T.-C.5
  • 12
    • 55549138762 scopus 로고    scopus 로고
    • A distributed SDP approach for large-scale noisy anchor-free graph reailzation with applications to molecular conformation
    • P. Biswas, K.-C. Toh, and Y. Ye, A distributed SDP approach for large-scale noisy anchor-free graph reailzation with applications to molecular conformation. SIAM J. Sci. Comput. 30 (2008), 1251-1277. Zbl 1161.49028 MR 2398864
    • (2008) SIAM J. Sci. Comput , vol.30 , pp. 1251-1277
    • Biswas, P.1    Toh, K.-C.2    Ye, Y.3
  • 14
    • 34547494898 scopus 로고    scopus 로고
    • A distributed method for solving semidefinite programs arising from ad hoc wireless sensor network localization
    • Springer New York
    • P. Biswas and Y. Ye, A distributed method for solving semidefinite programs arising from ad hoc wireless sensor network localization. In Multiscale optimization methods and applications, Nonconvex Optim. Appl. 82, Springer, New York 2006, 69-84. Zbl 1100.90029 MR 2191577
    • (2006) In Multiscale optimization methods and applications, Nonconvex Optim. Appl , vol.82 , pp. 69-84
    • Biswas, P.1    Ye, Y.2
  • 17
    • 84877145514 scopus 로고
    • Theory and applications of distance geometry
    • New York
    • L. M. Blumenthal, Theory and applications of distance geometry. Second edition, Chelsea Publishing Co., New York 1970. Zbl 0208.24801 MR 0268781
    • (1970) Second edition, Chelsea Publishing Co
    • Blumenthal, L.M.1
  • 18
    • 0032910450 scopus 로고    scopus 로고
    • Three-dimensional analysis of craniofacial form in a familial rabbit model of nonsyndromic coronal suture synostosis using Euclidean distance matrix analysis
    • A. M. Burrows, J. T. Richtsmeier, M. P. Mooney, T. D. Smith, H. W. Losken, and M. I. Siegel, Three-dimensional analysis of craniofacial form in a familial rabbit model of nonsyndromic coronal suture synostosis using Euclidean distance matrix analysis. The Cleft Palate-Craniofacial Journal 36 (1999), 196-206.
    • (1999) The Cleft Palate-Craniofacial Journal , vol.36 , pp. 196-206
    • Burrows, A.M.1    Richtsmeier, J.T.2    Mooney, M.P.3    Smith, T.D.4    Losken, H.W.5    Siegel, M.I.6
  • 19
    • 85032750937 scopus 로고    scopus 로고
    • An introduction to compressive sensing
    • E. Candes and M. Wakin, An introduction to compressive sensing. IEEE Signal Process. Mag. 25 (2008), no. 2, 21-30.
    • (2008) IEEE Signal Process. Mag , vol.25 , Issue.2 , pp. 21-30
    • Candes, E.1    Wakin, M.2
  • 20
    • 34547494277 scopus 로고    scopus 로고
    • SpaseLoc: an adaptive subproblem algorithm for scalable wireless sensor network localization
    • M. W. Carter, H. H. Jin, M. A. Saunders, and Y. Ye, SpaseLoc: an adaptive sub- problem algorithm for scalable wireless sensor network localization. SIAM J. Optim. 17 (2006), 1102-1128. Zbl 1136.90321 MR 2274505
    • (2006) SIAM J. Optim , vol.17 , pp. 1102-1128
    • Carter, M.W.1    Jin, H.H.2    Saunders, M.A.3    Ye, Y.4
  • 23
    • 0002860073 scopus 로고
    • Rigidity and energy
    • R. Connelly, Rigidity and energy. Invent. Math. 66 (1982), 11-33. Zbl 0485.52001 MR 652643
    • (1982) Invent. Math , vol.66 , pp. 11-33
    • Connelly, R.1
  • 24
    • 4243199684 scopus 로고
    • On generic global rigidity, In Applied geometry and discrete mathematics, DIMACS Ser
    • Amer. Math. Soc., Providence, RI
    • R. Connelly, On generic global rigidity. In Applied geometry and discrete mathe- matics, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 4, Amer. Math. Soc., Providence, RI, 1991, 147-155. Zbl 0736.05069 MR 1116345
    • (1991) Discrete Math. Theoret. Comput. Sci , vol.4 , pp. 147-155
    • Connelly, R.1
  • 25
    • 0002909309 scopus 로고
    • Rigidity
    • North-Holland, Amsterdam
    • R. Connelly, Rigidity. In Handbook of convex geometry. Vol. A, North-Holland, Amsterdam 1993, 223-271. Zbl 0788.52001 MR 1242981
    • (1993) Handbook of convex geometry , vol.A , pp. 223-271
    • Connelly, R.1
  • 26
    • 0012166973 scopus 로고    scopus 로고
    • Tensegrity frameworks: Why are they stable? In Rigidity theory and applications, Kluwer Academic/Plenum Publishers
    • R. Connelly, Tensegrity frameworks: Why are they stable? In Rigidity theory and applications, Kluwer Academic/Plenum Publishers, New York, 1999, 47-54.
    • (1999) New York , pp. 47-54
    • Connelly, R.1
  • 27
    • 17444406687 scopus 로고    scopus 로고
    • Generic global rigidity
    • R. Connelly, Generic global rigidity. Discrete Comput. Geom. 33 (2005), 549-563. Zbl 1072.52016 MR 2132290
    • (2005) Discrete Comput. Geom , vol.33 , pp. 549-563
    • Connelly, R.1
  • 28
    • 33244484959 scopus 로고    scopus 로고
    • Technical report, Dept. of Math., Cornell University, Ithaca, NY
    • R. Connelly and M. Sloughter, Realizability of graphs. Technical report, Dept. of Math., Cornell University, Ithaca, NY, 2004.
    • (2004) Realizability of graphs
    • Connelly, R.1    Sloughter, M.2
  • 29
    • 33745240949 scopus 로고    scopus 로고
    • Distributed weighted-multidimensional scaling for node localization in sensor networks
    • J. A. Costa, N. Patwari, and A. O. Hero, III, Distributed weighted-multidimensional scaling for node localization in sensor networks. ACM Trans. Sensor Network 2 (2006), 39-64.
    • (2006) ACM Trans. Sensor Network , vol.2 , pp. 39-64
    • Costa, J.A.1    Patwari, N.2    Hero III, A.O.3
  • 30
    • 0011962258 scopus 로고
    • Chemical distance geometry: Current realization and future projection
    • G. M. Crippen, Chemical distance geometry: Current realization and future projec- tion. J. Math. Chem. 6 (1991), 307-324.
    • (1991) J. Math. Chem , vol.6 , pp. 307-324
    • Crippen, G.M.1
  • 32
    • 35048875877 scopus 로고    scopus 로고
    • On the automation of similarity information maintenance in flexible query answering systems, Database and expert systems applications (DEXA 2004)
    • Springer-Verlag Berlin
    • B. C. Csáji, J. Küng, J. Palkoska and R. Wagner. On the automation of similarity information maintenance in flexible query answering systems. In Database and expert systems applications (DEXA 2004), Lect. Notes Comput. Sci. 3180, Springer-Verlag, Berlin 2004, 130-140.
    • (2004) Lect. Notes Comput. Sci , vol.3180 , pp. 130-140
    • Csáji, B.C.1    Küng, J.2    Palkoska, J.3    Wagner, R.4
  • 34
    • 41549152026 scopus 로고    scopus 로고
    • Equality relating Euclidean distance cone to positive semidefinite cone
    • J. Dattorro, Equality relating Euclidean distance cone to positive semidefinite cone. Linear Algebra Appl. 428 (2008), 2597-2600. Zbl 1142.15014 MR 2416574
    • (2008) Linear Algebra Appl , vol.428 , pp. 2597-2600
    • Dattorro, J.1
  • 36
    • 77949422811 scopus 로고    scopus 로고
    • Sensor network localization, Euclidean distance matrix completions, and graph realization
    • Y. Ding, N. Krislock, J. Qian, and H. Wolkowicz, Sensor network localization, Euclidean distance matrix completions, and graph realization. Optim. Eng. 11 (2010), 45-66. MR 2601732
    • (2010) Optim. Eng , vol.11 , pp. 45-66
    • Ding, Y.1    Krislock, N.2    Qian, J.3    Wolkowicz, H.4
  • 37
    • 0035013232 scopus 로고    scopus 로고
    • Convex position estimation in wireless sensor networks
    • IEEE, Piscataway NJ
    • L. Doherty, K. S. J. Pister, and L. El Ghaoui, Convex position estimation in wireless sensor networks. In Proceedings IEEE INFOCOM 2001, Vol. 3, IEEE, Piscataway, NJ, 2001, 1655-1663.
    • (2001) Proceedings IEEE INFOCOM , vol.2001 , pp. 1655-1663
    • Doherty, L.1    Pister, K.S.J.2    El Ghaoui, L.3
  • 38
    • 84867962761 scopus 로고    scopus 로고
    • A geometric build-up algorithm for solving the molecular distance geometry problem with sparse distance data
    • Q. Dong and Z. Wu, A geometric build-up algorithm for solving the molecular distance geometry problem with sparse distance data. J. Global Optim. 26 (2003), 321-333. Zbl 1032.92036 MR 1977953
    • (2003) J. Global Optim , vol.26 , pp. 321-333
    • Dong, Q.1    Wu, Z.2
  • 41
    • 0041528360 scopus 로고
    • Three theorems with applications to Euclidean distance matrices
    • R. W. Farebrother, Three theorems with applications to Euclidean distance matrices. Linear Algebra Appl. 95 (1987), 11-16. Zbl 0627.15008 MR 907390
    • (1987) Linear Algebra Appl , vol.95 , pp. 11-16
    • Farebrother, R.W.1
  • 42
    • 0028931493 scopus 로고
    • A three-dimensional evaluation of human facial asymmetry
    • V. F. Ferrario, C. Sforza, A. Miani, Jr, and G. Serrao, A three-dimensional evalua- tion of human facial asymmetry. J. Anat. 186 (1995), 103-110.
    • (1995) J. Anat , vol.186 , pp. 103-110
    • Ferrario, V.F.1    Sforza, C.2    Miani Jr, A.3    Serrao, G.4
  • 44
    • 0026622712 scopus 로고
    • Forests, frames, and games: algorithms for matroid sums and applications
    • H. N. Gabow and H. H. Westermann, Forests, frames, and games: algorithms for matroid sums and applications. Algorithmica 7 (1992), 465-497. Zbl 0771.05026 MR 1154585
    • (1992) Algorithmica , vol.7 , pp. 465-497
    • Gabow, H.N.1    Westermann, H.H.2
  • 45
    • 0002670267 scopus 로고
    • Neighboring vertices on a convex polyhedron
    • Princeton University Press, Princeton N. J
    • D. Gale, Neighboring vertices on a convex polyhedron. In Linear inequalities and related systems, Ann. of Math. Stud. 38, Princeton University Press, Princeton, N.J., 1956, 255-263. Zbl 0072.37805 MR 0085552
    • (1956) Linear inequalities and related systems, Ann. of Math. Stud , vol.38 , pp. 255-263
    • Gale, D.1
  • 48
    • 0000771519 scopus 로고
    • An alternating projection algorithm for computing the nearest Euclidean distance matrix
    • W. Glunt, T. L. Hayden, S. Hong, and J. Wells, An alternating projection algorithm for computing the nearest Euclidean distance matrix. SIAM J. Matrix Anal. Appl. 11 (1990), 589-600. Zbl 0728.65034 MR 1066161
    • (1990) SIAM J. Matrix Anal. Appl , vol.11 , pp. 589-600
    • Glunt, W.1    Hayden, T.L.2    Hong, S.3    Wells, J.4
  • 49
    • 77955871459 scopus 로고    scopus 로고
    • Characterizing generic global rigidity
    • S. J. Gortler, A. D. Healy, and D. P. Thurston, Characterizing generic global rigid- ity. Amer. J. Math. 132 (2010), 897-939. Zbl 1202.52020 MR 2663644
    • (2010) Amer. J. Math , vol.132 , pp. 897-939
    • Gortler, S.J.1    Healy, A.D.2    Thurston, D.P.3
  • 51
    • 0009245040 scopus 로고
    • Euclidean distance geometry
    • J. C. Gower, Euclidean distance geometry. Math. Sci. 7 (1982), 1-14. Zbl 0492.51017 MR 642165
    • (1982) Math. Sci , vol.7 , pp. 1-14
    • Gower, J.C.1
  • 52
    • 0038087036 scopus 로고
    • Properties of Euclidean and non-Euclidean distance matrices
    • J. C. Gower, Properties of Euclidean and non-Euclidean distance matrices. Linear Algebra Appl. 67 (1985), 81-97. Zbl 0569.15016 MR 787367
    • (1985) Linear Algebra Appl , vol.67 , pp. 81-97
    • Gower, J.C.1
  • 53
    • 35348959185 scopus 로고    scopus 로고
    • History of Mathematics 19, American Mathematical Society, Providence, RI
    • H. Grassmann, Extension theory. History of Mathematics 19, American Mathemati- cal Society, Providence, RI, 2000. Zbl 0953.01025 MR 1747519
    • (2000) Extension theory
    • Grassmann, H.1
  • 55
    • 51249185617 scopus 로고
    • The ellipsoid method and its consequences in combinatorial optimization
    • M. Grötschel, L. Lovász, and A. Schrijver, The ellipsoid method and its conse- quences in combinatorial optimization. Combinatorica 1 (1981), 169-197. Zbl 0492.90056 MR 625550
    • (1981) Combinatorica , vol.1 , pp. 169-197
    • Grötschel, M.1    Lovász, L.2    Schrijver, A.3
  • 59
    • 0026821495 scopus 로고
    • Conditions for unique graph realizations
    • B. Hendrickson, Conditions for unique graph realizations. SIAM J. Comput. 21 (1992), 65-84. Zbl 0756.05047 MR 1148818
    • (1992) SIAM J. Comput , vol.21 , pp. 65-84
    • Hendrickson, B.1
  • 60
    • 0013558430 scopus 로고
    • The molecule problem: exploiting structure in global optimization
    • B. Hendrickson, The molecule problem: exploiting structure in global optimization. SIAM J. Optim. 5 (1995), 835-857. Zbl 0844.05093 MR 1358807
    • (1995) SIAM J. Optim , vol.5 , pp. 835-857
    • Hendrickson, B.1
  • 61
    • 84867966586 scopus 로고    scopus 로고
    • Some properties for the Euclidean distance matrix and positive semidefinite matrix completion problems
    • H.-X. Huang, Z.-A. Liang, and P. M. Pardalos, Some properties for the Euclidean distance matrix and positive semidefinite matrix completion problems. J. Global Optim. 25 (2003), 3-21. Zbl 1037.15015 MR 1969424
    • (2003) J. Global Optim , vol.25 , pp. 3-21
    • Huang, H.-X.1    Liang, Z.-A.2    Pardalos, P.M.3
  • 62
    • 15444370051 scopus 로고    scopus 로고
    • Connected rigidity matroids and unique realizations of graphs
    • B. Jackson and T. Jordán, Connected rigidity matroids and unique realizations of graphs. J. Combin. Theory Ser. B 94 (2005), 1-29. Zbl 1076.05021 MR 2130278
    • (2005) J. Combin. Theory Ser. B , vol.94 , pp. 1-29
    • Jackson, B.1    Jordán, T.2
  • 63
    • 32344449818 scopus 로고    scopus 로고
    • Globally linked pairs of vertices in equivalent realizations of graphs
    • B. Jackson, T. Jordán, and Z. Szabadka, Globally linked pairs of vertices in equi- valent realizations of graphs. Discrete Comput. Geom. 35 (2006), 493-512. Zbl 1088.05056 MR 2202116
    • (2006) Discrete Comput. Geom , vol.35 , pp. 493-512
    • Jackson, B.1    Jordán, T.2    Szabadka, Z.3
  • 65
    • 0031999150 scopus 로고    scopus 로고
    • An interior-point method for approximate positive semidefinite completions
    • C. R. Johnson, B. Kroschel, and H. Wolkowicz, An interior-point method for approximate positive semidefinite completions. Comput. Optim. Appl. 9 (1998), 175-190. Zbl 0907.90207 MR 1601892
    • (1998) Comput. Optim. Appl , vol.9 , pp. 175-190
    • Johnson, C.R.1    Kroschel, B.2    Wolkowicz, H.3
  • 66
    • 21844491255 scopus 로고
    • Connections between the real positive semidefinite and distance matrix completion problems
    • C. R. Johnson and P. Tarazaga, Connections between the real positive semidefinite and distance matrix completion problems. Linear Algebra Appl. 223/224 (1995), 375-391. Zbl 0827.15032 MR 1340702
    • (1995) Linear Algebra Appl , vol.223 , Issue.224 , pp. 375-391
    • Johnson, C.R.1    Tarazaga, P.2
  • 68
    • 70450210946 scopus 로고    scopus 로고
    • Exploiting sparsity in SDP relaxation for sensor network localization
    • S. Kim, M. Kojima, and H. Waki, Exploiting sparsity in SDP relaxation for sensor network localization. SIAM J. Optim. 20 (2009), 192-215. Zbl 1190.65096 MR 2496899
    • (2009) SIAM J. Optim , vol.20 , pp. 192-215
    • Kim, S.1    Kojima, M.2    Waki, H.3
  • 73
    • 77956091432 scopus 로고    scopus 로고
    • Explicit sensor network localization using semidefinite representations and facial reductions
    • N. Krislock and H. Wolkowicz, Explicit sensor network localization using semi- definite representations and facial reductions. SIAM J. Optim. 20 (2010), 2679- 2708. MR 2678410
    • (2010) SIAM J. Optim , vol.20 , pp. 2679-2708
    • Krislock, N.1    Wolkowicz, H.2
  • 74
    • 0014856102 scopus 로고
    • On graphs and rigidity of plane skeletal structures
    • G. Laman, On graphs and rigidity of plane skeletal structures. J. Engrg. Math. 4 (1970), 331-340. Zbl 0213.51903 MR 0269535
    • (1970) J. Engrg. Math , vol.4 , pp. 331-340
    • Laman, G.1
  • 75
    • 0035238864 scopus 로고    scopus 로고
    • Global optimization with polynomials and the problem of moments
    • J. B. Lasserre, Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11 (2001), 796-817. Zbl 1010.90061 MR 1814045
    • (2001) SIAM J. Optim , vol.11 , pp. 796-817
    • Lasserre, J.B.1
  • 76
    • 0000160366 scopus 로고    scopus 로고
    • Cuts, matrix completions and graph rigidity
    • M. Laurent, Cuts, matrix completions and graph rigidity. Math. Programming 79 (1997), 255-283. Zbl 0887.90174 MR 1464770
    • (1997) Math. Programming , vol.79 , pp. 255-283
    • Laurent, M.1
  • 77
    • 0007302409 scopus 로고    scopus 로고
    • A tour d'horizon on positive semidefinite and Euclidean distance matrix completion problems
    • Amer. Math. Soc., Providence, RI
    • M. Laurent, A tour d'horizon on positive semidefinite and Euclidean distance matrix completion problems. In Topics in semidefinite and interior-point methods, Fields Inst. Commun. 18, Amer. Math. Soc., Providence, RI, 1998, 51-76. Zbl 0906.05074 MR 1607310
    • (1998) Topics in semidefinite and interior-point methods, Fields Inst. Commun , vol.18 , pp. 51-76
    • Laurent, M.1
  • 80
    • 0031285908 scopus 로고    scopus 로고
    • Global continuation for distance geometry problems
    • J. J. Moré and Z. Wu, Global continuation for distance geometry problems. SIAM J. Optim. 7 (1997), 814-836. Zbl 0891.90168 MR 1462067
    • (1997) SIAM J. Optim , vol.7 , pp. 814-836
    • Moré, J.J.1    Wu, Z.2
  • 81
    • 0000935435 scopus 로고    scopus 로고
    • Distance geometry optimization for protein structures
    • J. J. Moré and Z. Wu, Distance geometry optimization for protein structures. J. Global Optim. 15 (1999), 219-234. Zbl 0944.92012 MR 1737437
    • (1999) J. Global Optim , vol.15 , pp. 219-234
    • Moré, J.J.1    Wu, Z.2
  • 84
    • 67349132549 scopus 로고    scopus 로고
    • Sum of squares method for sensor network localization
    • J. Nie, Sum of squares method for sensor network localization. Comput. Optim. Appl. 43 (2009), 151-179. Zbl 1170.90510 MR 2506248
    • (2009) Comput. Optim. Appl , vol.43 , pp. 151-179
    • Nie, J.1
  • 87
    • 0031285890 scopus 로고    scopus 로고
    • Strong duality for semidefinite programming
    • M. V. Ramana, L. Tunçel, and H. Wolkowicz, Strong duality for semidefinite pro- gramming. SIAM J. Optim. 7 (1997), 641-662. Zbl 0891.90129 MR 1462059
    • (1997) SIAM J. Optim , vol.7 , pp. 641-662
    • Ramana, M.V.1    Tunçel, L.2    Wolkowicz, H.3
  • 88
    • 78549288866 scopus 로고    scopus 로고
    • Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization
    • B. Recht, M. Fazel, and P. A. Parrilo, Guaranteed minimum-rank solutions of lin- ear matrix equations via nuclear norm minimization. SIAM Rev. 52 (2010), 471-501. Zbl 1198.90321 MR 2680543
    • (2010) SIAM Rev , vol.52 , pp. 471-501
    • Recht, B.1    Fazel, M.2    Parrilo, P.A.3
  • 92
    • 0002702652 scopus 로고
    • Embeddability of weighted graphs in k-space is strongly NP-hard
    • Control, and Computing, University of Illinois, Department of Electrical Engineering, UrbanaChampaign, Illinois
    • J. B. Saxe, Embeddability of weighted graphs in k-space is strongly NP-hard. In Pro- ceedings Seventeenth Annual Allerton Conference on Communication, Control, and Computing, University of Illinois, Department of Electrical Engineering, Urbana- Champaign, Illinois, 1979.
    • (1979) Proceedings Seventeenth Annual Allerton Conference on Communication
    • Saxe, J.B.1
  • 93
    • 0001203499 scopus 로고
    • Remarks to Maurice Fréchet's article Sur la définition axiomatique d'une classe d'espace distanciés vectoriellement applicable sur l'espace de Hilbert
    • I. J. Schoenberg, Remarks to Maurice Fréchet's article "Sur la définition axioma- tique d'une classe d'espace distanciés vectoriellement applicable sur l'espace de Hilbert". Ann. of Math. (2) 36 (1935), 724-732. JFM 61.0435.04 Zbl 0012.30703 MR 1503248
    • (1935) Ann. of Math , Issue.2 , pp. 724-732
    • Schoenberg, I.J.1
  • 94
    • 33846661142 scopus 로고    scopus 로고
    • Theory of semidefinite programming for sensor network localization
    • A. M.-C. So and Y. Ye, Theory of semidefinite programming for sensor network localization. Math. Programming 109 (2007), 367-384. Zbl 05131081 MR 2295148
    • (2007) Math. Programming , vol.109 , pp. 367-384
    • So, A.-C.1    Ye, Y.2
  • 97
    • 0002294618 scopus 로고    scopus 로고
    • Applications of multidimensional scaling to molecular conformation
    • M. W. Trosset, Applications of multidimensional scaling to molecular conformation. Computing Science and Statistics 29 (1998), 148-152.
    • (1998) Computing Science and Statistics , vol.29 , pp. 148-152
    • Trosset, M.W.1
  • 98
    • 0034296096 scopus 로고    scopus 로고
    • Distance matrix completion by numerical optimization
    • M. W. Trosset, Distance matrix completion by numerical optimization. Comput. Optim. Appl. 17 (2000), 11-22. Zbl 0964.65047 MR 1791595
    • (2000) Comput. Optim. Appl , vol.17 , pp. 11-22
    • Trosset, M.W.1
  • 99
    • 39449085876 scopus 로고    scopus 로고
    • Second-order cone programming relaxation of sensor network localization
    • P. Tseng, Second-order cone programming relaxation of sensor network localization. SIAM J. Optim. 18 (2007), 156-185. Zbl 1176.90454 MR 2299679
    • (2007) SIAM J. Optim , vol.18 , pp. 156-185
    • Tseng, P.1
  • 101
    • 33847158335 scopus 로고    scopus 로고
    • Sums of squares and semidefinite program relaxations for polynomial optimization problems with structured sparsity
    • H. Waki, S. Kim, M. Kojima, and M. Muramatsu, Sums of squares and semidefinite program relaxations for polynomial optimization problems with structured sparsity. SIAM J. Optim. 17 (2006), 218-242. Zbl 1109.65058 MR 2219151
    • (2006) SIAM J. Optim , vol.17 , pp. 218-242
    • Waki, H.1    Kim, S.2    Kojima, M.3    Muramatsu, M.4
  • 102
    • 60549085200 scopus 로고    scopus 로고
    • Further relaxations of the semidefinite programming approach to sensor network localization
    • Z. Wang, S. Zheng, Y. Ye, and S. Boyd, Further relaxations of the semidefinite programming approach to sensor network localization. SIAM J. Optim. 19 (2008), 655-673. Zbl 1173.90498 MR 2425034
    • (2008) SIAM J. Optim , vol.19 , pp. 655-673
    • Wang, Z.1    Zheng, S.2    Ye, Y.3    Boyd, S.4
  • 104
    • 0038024955 scopus 로고
    • Matroids and rigid structures
    • Cambridge Univ. Press, Cambridge
    • W. Whiteley, Matroids and rigid structures. In Matroid applications, Encyclopedia Math. Appl. 40, Cambridge Univ. Press, Cambridge 1992, 1-53. Zbl 0768.05025 MR 1165538
    • (1992) In Matroid applications, Encyclopedia Math. Appl , vol.40 , pp. 1-53
    • Whiteley, W.1
  • 106
    • 33847251002 scopus 로고    scopus 로고
    • An updated geometric build-up algorithm for solving the molecular distance geometry problems with sparse distance data
    • D. Wu and Z. Wu, An updated geometric build-up algorithm for solving the mo- lecular distance geometry problems with sparse distance data. J. Global Optim. 37 (2007), 661-673. Zbl 1119.92032 MR 2302995
    • (2007) J. Global Optim , vol.37 , pp. 661-673
    • Wu, D.1    Wu, Z.2
  • 108
    • 84877106998 scopus 로고
    • On the best Euclidean fit to a distance matrix
    • F. Z. Zhang, On the best Euclidean fit to a distance matrix. J. Beijing Normal Univ. (Nat. Sci.) 1987 (1987), No. 4, 21-24. Zbl 0655.15023 MR 939410
    • (1987) J. Beijing Normal Univ. (Nat. Sci) , vol.1987 , Issue.4 , pp. 21-24
    • Zhang, F.Z.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.