Skip to main content
SpringerLink
Log in
Book cover

Handbook on Semidefinite, Conic and Polynomial Optimization pp 879–914Cite as

Euclidean Distance Matrices and Applications

  • Chapter
  • First Online:

  • 6246 Accesses

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 166))

Abstract

Euclidean distance matrices, or EDMs, have been receiving increased attention for two main reasons. The first reason is that the many applications of EDMs, such as molecular conformation in bioinformatics, dimensionality reduction in machine learning and statistics, and especially the problem of wireless sensor network localization, have all become very active areas of research. The second reason for this increased interest is the close connection between EDMs and semidefinite matrices. Our recent ability to solve semidefinite programs efficiently means we can now also solve many problems involving EDMs efficiently. This chapter connects the classical approaches for EDMs with the more recent tools from semidefinite programming. We emphasize the application to sensor network localization.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   219.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   279.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   279.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    Early appearances of this linear operator are in [104, 119]. However, the use of the notation \(\mathcal{K}\) for this linear operator dates back to [43] wherein κ was used due to the fact that the formula for \(\mathcal{K}\) is basically the cosine law (\({c}^{2} = {a}^{2} + {b}^{2} - 2ab\cos (\gamma )\)). Later on, in [78], K was used to denote this linear operator.

References

  1. Akyildiz, I.F., Su, W., Sankarasubramaniam, Y., Cayirci, E.: Wireless sensor networks: a survey. Comput. Netw. 38, 393–422 (2002)

    Google Scholar 

  2. Al-Homidan, S., Wolkowicz, H.: Approximate and exact completion problems for Euclidean distance matrices using semidefinite programming. Linear Algebra Appl. 406, 109–141 (2005)

    Google Scholar 

  3. Alfakih, A.Y., Anjos, M.F., Piccialli, V., Wolkowicz, H.: Euclidean distance matrices, semidefinite programming, and sensor network localization. Port.Math. 68, 53–102 (2011)

    Google Scholar 

  4. Alfakih, A.Y., Khandani, A., Wolkowicz, H.: Solving Euclidean distance matrix completion problems via semidefinite programming. Comput. Optim. Appl. 12, 13–30 (1999)

    Google Scholar 

  5. Alfakih, A.Y., Wolkowicz, H.: Matrix completion problems. In: Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds.) Handbook of Semidefinite Programming: Theory, Algorithms, and Applications. Int. Ser. Oper. Res. Man. Sc., vol. 27, pp. 533–545. Kluwer Academic, Boston (2000)

    Google Scholar 

  6. Alfakih, A.Y.: Graph rigidity via Euclidean distance matrices. Linear Algebra Appl. 310, 149–165 (2000)

    Google Scholar 

  7. Alfakih, A.Y.: On rigidity and realizability of weighted graphs. Linear Algebra Appl. 325, 57–70 (2001)

    Google Scholar 

  8. Alfakih, A.Y.: On the uniqueness of Euclidean distance matrix completions. Linear Algebra Appl. 370, 1–14 (2003)

    Google Scholar 

  9. Alfakih, A.Y.: On the uniqueness of Euclidean distance matrix completions: the case of points in general position. Linear Algebra Appl. 397, 265–277 (2005)

    Google Scholar 

  10. Alfakih, A.Y.: On the nullspace, the rangespace and the characteristic polynomial of Euclidean distance matrices. Linear Algebra Appl. 416, 348–354 (2006)

    Google Scholar 

  11. Alfakih, A.Y.: A remark on the faces of the cone of Euclidean distance matrices. Linear Algebra Appl. 414, 266–270 (2006)

    Google Scholar 

  12. Alfakih, A.Y.: On dimensional rigidity of bar-and-joint frameworks. Discrete Appl. Math. 155, 1244–1253 (2007)

    Google Scholar 

  13. Alfakih, A.Y., Wolkowicz, H.: Two theorems on Euclidean distance matrices and Gale transform. Linear Algebra Appl. 340, 149–154 (2002)

    Google Scholar 

  14. Ames, B.P.W., Vavasis, S.A.: Nuclear norm minimization for the planted clique and biclique problems. http://arxiv.org/abs/0901.3348 (2009). Accessed 21 Jan 2009

  15. An, L.T.H., Tao, P.D.: Large-scale molecular optimization from distance matrices by a D.C. optimization approach. SIAM J. Optim. 14, 77–114 (2003)

    Google Scholar 

  16. Aspnes, J., Eren, T., Goldenberg, D.K., Morse, A., Whiteley, W., Yang, Y.R., Anderson, B.D.O., Belhumeur, P.N.: A theory of network localization. IEEE T. Mobile Comput. 5, 1663–1678 (2006)

    Google Scholar 

  17. Aspnes, J., Goldenberg, D., Yang, Y.R.: On the computational complexity of sensor network localization. Lect. Notes Comput. Sc. 3121, 32–44 (2004)

    Google Scholar 

  18. Bădoiu, M., Demaine, E.D., Hajiaghayi, M., Indyk, P.: Low-dimensional embedding with extra information. Discrete Comput. Geom. 36, 609–632 (2006)

    Google Scholar 

  19. Bakonyi, M., Johnson, C.: The Euclidean distance matrix completion problem, SIAM J. Matrix Anal. Appl. 16, 646–654 (1995)

    Google Scholar 

  20. Barvinok, A.: Problems of distance geometry and convex properties of quadratic maps. Discrete Comput. Geom. 13, 189–202 (1995)

    Google Scholar 

  21. Beck, A., Stoica, P., Li, J.: Exact and approximate solutions of source localization problems. IEEE T. Signal Proces. 56, 1770–1778 (2008)

    Google Scholar 

  22. Beck, A., Teboulle, M., Chikishev, Z.: Iterative minimization schemes for solving the single source localization problem. SIAM J. Optim. 19, 1397–1416 (2008)

    Google Scholar 

  23. Belk, M.: Realizability of graphs in three dimensions. Discrete Comput. Geom. 37, 139–162 (2007)

    Google Scholar 

  24. Belk, M., Connelly, R.: Realizability of graphs. Discrete Comput. Geom., 37, 125–137 (2007)

    Google Scholar 

  25. Biswas, P.: Semidefinite programming approaches to distance geometry problems. Ph.D. thesis, Stanford University (2007)

    Google Scholar 

  26. Biswas, P., Liang, T.-C., Toh, K.-C., Wang, T.-C., Ye, Y.: Semidefinite programming approaches for sensor network localization with noisy distance measurements, IEEE T. Autom. Sci. Eng. 3, 360–371 (2006)

    Google Scholar 

  27. Biswas, P., Toh, K.-C., Ye, Y.: A distributed SDP approach for large-scale noisy anchor-free graph realization with applications to molecular conformation. SIAM J. Sci. Comput. 30, 1251–1277 (2008)

    Google Scholar 

  28. Biswas, P., Ye, Y.: Semidefinite programming for ad hoc wireless sensor network localization. In: Information Processing in Sensor Networks, Berkeley, CA, 26–27 April 2004. Proceedings of the 3rd International Symposium on Information Processing in Sensor Networks, pp. 46–54. ACM, New York (2004)

    Google Scholar 

  29. Biswas, P., Ye, Y.: A distributed method for solving semidefinite programs arising from ad hoc wireless sensor network localization. In: Hager, W.W., Huang, S.-J., Pardalos, P.M., Prokopyev, O.A. (eds.) Multiscale Optimization Methods and Applications. Nonconvex Optim. Appl., vol. 82, pp. 69–84. Springer, New York (2006)

    Google Scholar 

  30. Björck, Å.: Numerical Methods for Least Squares Problems. SIAM, Philadelphia (1996)

    Google Scholar 

  31. Blumenthal, L.M.: Theory and Applications of Distance Geometry, 2nd edn. Chelsea, New York (1970)

    Google Scholar 

  32. Bruck, J., Gao, J., Jiang, A.: Localization and routing in sensor networks by local angle information. ACM Trans. Sen. Netw. 5, 1–31 (2009)

    Google Scholar 

  33. Bulusu, N., Heidemann, J., Estrin, D.: GPS-less low-cost outdoor localization for very small devices. IEEE Pers. Commun. 7, 28–34 (2000)

    Google Scholar 

  34. Candès, E.J., Plan, Y.: Matrix completion with noise. P. IEEE 98, 925–936 (2010)

    Google Scholar 

  35. Candès, E.J., Recht, B.: Exact matrix completion via convex optimization. Found. Comput. Math. 9, 717–772 (2009)

    Google Scholar 

  36. Carter, M.W., Jin, H.H., Saunders, M.A., Ye, Y.: SpaseLoc: An adaptive subproblem algorithm for scalable wireless sensor network localization. SIAM J. Optim. 17, 1102–1128 (2006)

    Google Scholar 

  37. Cassioli, A.: Solving the sensor network localization problem using an heuristic multistage approach. http://www.optimization-online.org/DB_HTML/2009/03/2267.html (2009). Accessed 13 Jan 2010

  38. Chua, C.B., Tunçel, L.: Invariance and efficiency of convex representations. Math. Program. 111, 113–140 (2008)

    Google Scholar 

  39. Connelly, R.: Generic global rigidity. Discrete Comput. Geom. 33, 549–563 (2005)

    Google Scholar 

  40. Costa, J.A., Patwari, N., Hero III, A.O.: Distributed weighted-multidimensional scaling for node localization in sensor networks. ACM Trans. Sen. Netw. 2, 39–64 (2006)

    Google Scholar 

  41. Crippen, G.M.: Chemical distance geometry: Current realization and future projection. J. Math. Chem. 6, 307–324 (1991)

    Google Scholar 

  42. Crippen, G.M., Havel, T.F.: Distance Geometry and Molecular Conformation. Chemometrics Series, vol. 15. Research Studies Press, Taunton (1988)

    Google Scholar 

  43. Critchley, F.: On certain linear mappings between inner-product and squared distance matrices. Linear Algebra Appl. 105, 91–107 (1988)

    Google Scholar 

  44. Dattorro, J.: Convex Optimization & Euclidean Distance Geometry. Meboo Publishing USA (2008)

    Google Scholar 

  45. Ding, Y., Krislock, N., Qian, J., Wolkowicz, H.: Sensor network localization, Euclidean distance matrix completions, and graph realization. In: MobiCom Annual International Conference on Mobile Computing and Networking, San Francisco, CA, 14–19 September 2008. Proceedings of the First ACM International Workshop on Mobile Entity Localization and Tracking in GPS-Less Environment, pp. 129–134. ACM, New York (2008)

    Google Scholar 

  46. Ding, Y., Krislock, N., Qian, J., Wolkowicz, H.: Sensor network localization, Euclidean distance matrix completions, and graph realization. Optim. Eng. 11, 45–66 (2010)

    Google Scholar 

  47. Doherty, L., Pister, K.S.J., El Ghaoui, L.: Convex position estimation in wireless sensor networks. In: IEEE INFOCOM, Anchorage, AK, 22–26 April 2001. Proceedings of the Twentieth Annual Joint Conference of the IEEE Computer and Communications Societies, vol. 3, pp. 1655–1663 (2001)

    Google Scholar 

  48. Dong, Q., Wu, Z.: A linear-time algorithm for solving the molecular distance geometry problem with exact inter-atomic distances. J. Global Optim. 22, 365–375 (2002)

    Google Scholar 

  49. Dong, Q., Wu, Z.: A geometric build-up algorithm for solving the molecular distance geometry problem with sparse distance data. J. Global Optim. 26, 321–333 (2003)

    Google Scholar 

  50. dos Santos Carvalho, R., Lavor, C., Protti, F.: Extending the geometric build-up algorithm for the molecular distance geometry problem. Inform. Process. Lett. 108, 234–237 (2008)

    Google Scholar 

  51. Eckart, C., Young, G.: The approximation of one matrix by another of lower rank. Psychometrika 1, 211–218 (1936)

    Google Scholar 

  52. Emiris, I.Z., Nikitopoulos, T.G.: Molecular conformation search by distance matrix perturbations. J. Math. Chem. 37, 233–253 (2005)

    Google Scholar 

  53. Eren, T., Goldenberg, O., Whiteley, W., Yang, Y.R., Morse, A.S., Anderson, B.D.O., Belhumeur, P.N.: Rigidity, computation, and randomization in network localization. In: IEEE INFOCOM, Hong Kong, 7–11 March 2004. Proceedings of the Twenty-third Annual Joint Conference of the IEEE Computer and Communications Societies, vol. 4, pp. 2673–2684 (2004)

    Google Scholar 

  54. Farebrother, R.W.: Three theorems with applications to Euclidean distance matrices. Linear Algebra Appl. 95, 11–16 (1987)

    Google Scholar 

  55. Fazel, M.: Matrix rank minimization with applications. Ph.D. thesis, Stanford University (2002)

    Google Scholar 

  56. Fazel, M., Hindi, H., Boyd, S.P.: Log-det heuristic for matrix rank minimization with applications to Hankel and Euclidean distance matrices. In: American Control Conference, Denver, CO, 4–6 June 2003. Proceedings of the 2003 American Control Conference, vol. 3, pp. 2156–2162 (2003)

    Google Scholar 

  57. Fukuda, M., Kojima, M., Murota, K., Nakata, K.: Exploiting sparsity in semidefinite programming via matrix completion I: General framework. SIAM J. Optim. 11, 647–674 (2001)

    Google Scholar 

  58. Glunt, W., Hayden, T., Raydan, M.: Molecular conformations from distance matrices. J. Comput. Chem. 14, 114–120 (1993)

    Google Scholar 

  59. Glunt, W., Hayden, T.L., Hong, S., Wells, J.: An alternating projection algorithm for computing the nearest Euclidean distance matrix. SIAM J. Matrix Anal. Appl. 11, 589–600 (1990)

    Google Scholar 

  60. Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145 (1995)

    Google Scholar 

  61. Goldfarb, D., Scheinberg, K.: Interior point trajectories in semidefinite programming. SIAM J. Optim. 8, 871–886 (1998)

    Google Scholar 

  62. Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)

    Google Scholar 

  63. Gower, J.C.: Euclidean distance geometry. Math. Sci. 7, 1–14 (1982)

    Google Scholar 

  64. Gower, J.C.: Distance matrices and their Euclidean approximation. In: Data Analysis and Informatics, Versailles, 4–7 October 1983. Proceedings of the Third International Symposium on Data Analysis and Informatics, pp. 3–21. North-Holland, Amsterdam, (1984)

    Google Scholar 

  65. Gower, J.C.: Properties of Euclidean and non-Euclidean distance matrices. Linear Algebra Appl. 67, 81–97 (1985)

    Google Scholar 

  66. Green, B.: The orthogonal approximation of an oblique structure in factor analysis. Psychometrika 17, 429–440 (1952)

    Google Scholar 

  67. Grone, R., Johnson, C.R., Sá, E.M., Wolkowicz, H.: Positive definite completions of partial Hermitian matrices. Linear Algebra Appl. 58, 109–124 (1984)

    Google Scholar 

  68. Grooms, I.G., Lewis, R.M., Trosset, M.W.: Molecular embedding via a second order dissimilarity parameterized approach. SIAM J. Sci. Comput. 31, 2733–2756 (2009)

    Google Scholar 

  69. Havel, T.F., Kuntz, I.D., Crippen, G.M.: The theory and practice of distance geometry. B. Math. Biol. 45, 665–720 (1983)

    Google Scholar 

  70. Havel, T.F.: Metric matrix embedding in protein structure calculations, NMR spectra analysis, and relaxation theory. Magn. Reson. Chem. 41, S37–S50 (2003)

    Google Scholar 

  71. Hayden, T.L., Wells, J., Liu, W.M., Tarazaga, P.: The cone of distance matrices. Linear Algebra Appl. 144, 153–169 (1991)

    Google Scholar 

  72. Hendrickson, B.: The molecule problem: Determining conformation from pairwise distances. Ph.D. thesis, Cornell University (1990)

    Google Scholar 

  73. Hendrickson, B.: Conditions for unique graph realizations. SIAM J. Comput. 21, 65–84 (1992)

    Google Scholar 

  74. Hendrickson, B.: The molecule problem: Exploiting structure in global optimization. SIAM J. Optim. 5, 835–857 (1995)

    Google Scholar 

  75. Higham, N.J.: Computing the polar decomposition—with applications. SIAM J. Sci. Stat. Comp. 7, 1160–1174 (1986)

    Google Scholar 

  76. Jackson, B., Jordán, T.: Connected rigidity matroids and unique realizations of graphs. J. Comb. Theory B 94, 1–29 (2005)

    Google Scholar 

  77. Jin, H.H.: Scalable sensor localization algorithms for wireless sensor networks. Ph.D. thesis, University of Toronto (2005)

    Google Scholar 

  78. Johnson, C.R., Tarazaga, P.: Connections between the real positive semidefinite and distance matrix completion problems. Linear Algebra Appl. 223/224, 375–391 (1995). Special issue honoring Miroslav Fiedler and Vlastimil Pták.

    Google Scholar 

  79. Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, IBM Thomas J. Watson Res. Center, Yorktown Heights, NY, 20–22 March 1972. Proceedings of a Symposium on the Complexity of Computer Computations, pp. 85–103. Plenum, New York (1972)

    Google Scholar 

  80. Kim, S., Kojima, M., Waki, H.: Exploiting sparsity in SDP relaxation for sensor network localization. SIAM J. Optim. 20, 192–215 (2009)

    Google Scholar 

  81. Kim, S., Kojima, M., Waki, H., Yamashita, M.: SFSDP: a sparse version of full semidefinite programming relaxation for sensor network localization problems. Research Report B-457, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology (2009)

    Google Scholar 

  82. Krislock, N.: Semidefinite facial reduction for low-rank euclidean distance matrix completion. Ph.D. thesis, University of Waterloo (2010)

    Google Scholar 

  83. Krislock, N., Wolkowicz, H.: Explicit sensor network localization using semidefinite representations and facial reductions. SIAM J. Optim. 20, 2679–2708 (2010)

    Google Scholar 

  84. Kumar, P.S., Madhavan, C.V.: Minimal vertex separators of chordal graphs. Discrete Appl. Math. 89, 155–168 (1998)

    Google Scholar 

  85. Laurent, M.: A connection between positive semidefinite and Euclidean distance matrix completion problems. Linear Algebra Appl. 273, 9–22 (1998)

    Google Scholar 

  86. Laurent, M.: A tour d’horizon on positive semidefinite and Euclidean distance matrix completion problems. In: Semidefinite Programming and Interior-Point Approaches for Combinatorial Optimization Problems, Fields Institute, Toronto, ON, 15–17 May 1996. Topics in Semidefinite and Interior-Point Methods (Fields Institute Communications), pp. 51–76. Amer. Math. Soc., Providence (1998)

    Google Scholar 

  87. Laurent, M.: Polynomial instances of the positive semidefinite and Euclidean distance matrix completion problems. SIAM J. Matrix Anal. Appl. 22, 874–894 (2001)

    Google Scholar 

  88. Leung, N.-H. Z., Kim-Chuan Toh, K.-C.: An SDP-based divide-and-conquer algorithm for large-scale noisy anchor-free graph realization. SIAM J. Sci. Comput. 31, 4351–4372 (2009)

    Google Scholar 

  89. Li, X.-Y.: Wireless Ad Hoc and Sensor Networks: Theory and Applications. Cambridge University Press (2008)

    Google Scholar 

  90. Megerian, S., Koushanfar, F., Potkonjak, M., Srivastava, M.B.: Worst and best-case coverage in sensor networks. IEEE T. Mobile Comput. 4, 84–92 (2005)

    Google Scholar 

  91. Moore, D., Leonard, J., Rus, D., Teller, S.: Robust distributed network localization with noisy range measurements. In: SenSys’04 ACM Conference on Embedded Network Sensor Systems, Baltimore, MD, 3–5 November 2004. Proceedings of the 2nd International Conference on Embedded Networked Sensor Systems, pp. 50–61. ACM, New York (2004)

    Google Scholar 

  92. Moré, J.J., Wu, Z.: Global continuation for distance geometry problems. SIAM J. Optim. 7, 814–836 (1997)

    Google Scholar 

  93. Moré, J.J., Wu, Z.: Distance geometry optimization for protein structures. J. Global Optim. 15, 219–234 (1999)

    Google Scholar 

  94. Nawaz, S.: Anchor Free Localization for Ad-hoc Wireless Sensor Networks. Ph.D. thesis, University of New South Wales (2008)

    Google Scholar 

  95. Nie, J.: Sum of squares method for sensor network localization. Comput. Optim. Appl. 43, 151–179 (2009)

    Google Scholar 

  96. Pong, T., Tseng, P.: (Robust) Edge-based semidefinite programming relaxation of sensor network localization. Math. Program. (2010). doi: 10.1007/s10107-009-0338-x

    Google Scholar 

  97. Ramana, M.V., Tunçel, L., Wolkowicz, H.: Strong duality for semidefinite programming. SIAM J. Optim. 7, 641–662 (1997)

    Google Scholar 

  98. Recht, B., Fazel, M., Parrilo, P.A.: Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52, 471–501 (2010)

    Google Scholar 

  99. Recht, B., Xu, W., Hassibi, B.: Necessary and sufficient conditions for success of the nuclear norm heuristic for rank minimization. http://arxiv.org/abs/0809.1260 (2008). Accessed 7 Sep 2008

  100. Rockafellar, R.T.: Convex Analysis. Princeton University Press (1970)

    Google Scholar 

  101. Savvides, A., Han, C.-C., Strivastava, M.B.: Dynamic fine-grained localization in ad-hoc networks of sensors. In: MobiCom’01, Rome, Italy, 16-21 July 2001. Proceedings of the 7th Annual International Conference on Mobile Computing and Networking, pp. 166–179. ACM, New York (2001)

    Google Scholar 

  102. Saxe, J.B.: Embeddability of weighted graphs in k-space is strongly NP-hard. In: Annual Allerton Conference on Communications, Control, and Computing, 10–12 October 1979. Proceedings of the 17th Allerton Conference on Communications, Control, and Computing, pp. 480–489 (1979)

    Google Scholar 

  103. Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press (1993)

    Google Scholar 

  104. Schoenberg, I.: Remarks to Maurice Fréchet’s article “Sur la définition axiomatique d’une classe d’espace distanciés vectoriellement applicable sur l’espace de Hilbert”. Ann. Math. 36, 724–732 (1935)

    Google Scholar 

  105. Schönemann, P.: A generalized solution of the orthogonal Procrustes problem. Psychometrika 31, 1–10 (1966)

    Google Scholar 

  106. So, A.M.-C.: A semidefinite programming approach to the graph realization problem: theory, applications and extensions. Ph.D. thesis, Stanford University (2007)

    Google Scholar 

  107. So, A.M.-C., Ye, Y.: A semidefinite programming approach to tensegrity theory and realizability of graphs. In: SODA’06, Miami, FL, 22–24 January 2006. Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 766–775. ACM, New York (2006)

    Google Scholar 

  108. So, A.M.-C., Ye, Y.: Theory of semidefinite programming for sensor network localization. Math. Program. 109, 367–384 (2007)

    Google Scholar 

  109. Stoyanova, T., Kerasiotis, F., Prayati, A., Papadopoulos, G.: Evaluation of impact factors on RSS accuracy for localization and tracking applications in sensor networks. Telecommun. Syst. 42, 235–248 (2009)

    Google Scholar 

  110. Tarazaga, P.: Faces of the cone of Euclidean distance matrices: characterizations, structure and induced geometry. Linear Algebra Appl. 408, 1–13 (2005)

    Google Scholar 

  111. Tarazaga, P., Hayden, T.L., Wells, J.: Circum-Euclidean distance matrices and faces. Linear Algebra Appl. 232, 77–96 (1996)

    Google Scholar 

  112. Tseng, P.: Second-order cone programming relaxation of sensor network localization. SIAM J. Optim. 18, 156–185 (2007)

    Google Scholar 

  113. Wang, Z., Zheng, S., Boyd, S., Ye, Y.: Further relaxations of the semidefinite programming approach to sensor network localization. SIAM J. Optim. 19, 655–673 (2008)

    Google Scholar 

  114. Weinberger, K.Q., Sha, F., Saul, L.K.: Learning a kernel matrix for nonlinear dimensionality reduction. In: ICML’04: Banff, AB, 4–8 July 2004. Proceedings of the 21st International Conference on Machine Learning, p. 106–113. ACM, New York (2004)

    Google Scholar 

  115. Wu, D., Wu, Z.: An updated geometric build-up algorithm for solving the molecular distance geometry problems with sparse distance data. J. Global Optim. 37, 661–673 (2007)

    Google Scholar 

  116. Wu, D., Wu, Z., Yuan, Y.: Rigid versus unique determination of protein structures with geometric buildup. Optim. Lett. 2, 319–331 (2008)

    Google Scholar 

  117. Yang, Z., Liu, Y., Li, X.-Y.: Beyond trilateration: On the localizability of wireless ad-hoc networks. In: IEEE INFOCOM, Rio de Janeiro, Brazil, 19–25 April 2009. INFOCOM 2009, IEEE, pp. 2392–2400 (2009)

    Google Scholar 

  118. Yemini, Y.: Some theoretical aspects of position-location problems. In: 20th Annual Symposium on Foundations of Computer Science, San Juan, Puerto Rico, 29–31 October 1979. 20th Annual Symposium on Foundations of Computer Science, pp. 1–8 (1979)

    Google Scholar 

  119. Young, G., Householder, A.: Discussion of a set of points in terms of their mutual distances. Psychometrika 3, 19–22 (1938)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON, Canada, N2L 3G1

    Nathan Krislock & Henry Wolkowicz

Authors
  1. Nathan Krislock
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Henry Wolkowicz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nathan Krislock .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Industrial, Ecole Polytechnique Montreal, Édouard-Montpetit Boulevard 2350, Montreal, QC, Canada, H3C 3A7

    Miguel F. Anjos

  2. LAAS-CNRS, Avenue du Colonel Roche 7, 31077, Toulouse Cedex 4, France

    Jean B. Lasserre

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Krislock, N., Wolkowicz, H. (2012). Euclidean Distance Matrices and Applications. In: Anjos, M.F., Lasserre, J.B. (eds) Handbook on Semidefinite, Conic and Polynomial Optimization. International Series in Operations Research & Management Science, vol 166. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0769-0_30

Download citation

Publish with us

Policies and ethics

Search

Navigation