Skip to main content
SpringerLink
Log in

Matrix Completion Problems

  • Reference work entry
Encyclopedia of Optimization

Matrix completion problems are concerned with determining whether partially specified matrices can be completed to fully specified matrices satisfying certain prescribed properties. In this article we survey some results and provide references about these problems for the following matrix properties: positive semidefinite matrices, Euclidean distance matrices, completely positive matrices, contraction matrices, and matrices of given rank. We treat mainly optimization and combinatorial aspects.

Introduction

A partial matrix is a matrix whose entries are specified only on a subset of its positions; a completion of a partial matrix is simply a specification of the unspecified entries. Matrix completion problemsare concerned with determining whether or not a completion of a partial matrix exists which satisfies some prescribed property. We consider here the following matrix properties: positive (semi) definite matrices, distance matrices, completely positive matrices,...

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

Access this chapter

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 1,699.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Agler, J., Helton, J. W., McCullough, S., and Rodman, L.: ‘Positive semidefinite matrices with a given sparsity pattern’, Linear Alg. & Its Appl.107 (1988), 101–149.

    MathSciNet  MATH  Google Scholar 

  2. Alfakih, A. Y., Khandani, A., and Wolkowicz, H.: ‘Solving Euclidean distance matrix completion problems via semidefinite programming’, Comput. Optim. Appl.12 (1998), 13–30.

    MathSciNet  Google Scholar 

  3. Alizadeh, F.: ‘Interior point methods in semidefinite programming with applications in combinatorial optimization’, SIAM J. Optim.5 (1995), 13–51.

    MathSciNet  MATH  Google Scholar 

  4. Bakonyi, M., and Johnson, C. R.: ‘The Euclidian distance matrix completion problem’, SIAM J. Matrix Anal. Appl.16 (1995), 646–654.

    MathSciNet  MATH  Google Scholar 

  5. Barahona, F., and Mahjoub, A. R.: ‘On the cut polytope’, Math. Program.36 (1986), 157–173.

    MathSciNet  MATH  Google Scholar 

  6. Barrett, W. W., Johnson, C. R., and Loewy, R.: ‘The real positive definite completion problem: cycle completability’, Memoirs Amer. Math. Soc.584 (1996).

    Google Scholar 

  7. Barrett, W. W., Johnson, C. R., and Lundquist, M.: ‘Determinantal formulas for matrix completions associated with chordal graphs’, Linear Alg. & Its Appl.121 (1989), 265–289.

    MathSciNet  MATH  Google Scholar 

  8. Barrett, W., Johnson, C. R., and Tarazaga, P.: ‘The real positive definite completion problem for a simple cycle’, Linear Alg. & Its Appl.192 (1993), 3–31.

    MathSciNet  MATH  Google Scholar 

  9. Barvinok, A. I.: ‘Feasibility testing for systems of real quadratic equations’, Discrete Comput. Geom.10 (1993), 1–13.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  11. Blumenthal, L. M.: Theory and applications of distance geometry, Oxford Univ. Press, 1953.

    Google Scholar 

  12. Cohen, N., Johnson, C. R., Rodman, L., and Woederman, H. J.: ‘Ranks of completions of partial matrices’, in H. Dym, et al. (eds.): The Gohberg Anniv. Coll., Vol. I, Birkhäuser, 1989 p. 165–185.

    Google Scholar 

  13. Crippen, G. M., and Havel, T. F.: Distance geometry and molecular conformation, Res. Studies Press 1988.

    Google Scholar 

  14. Deza, M., and Laurent, M.: Geometry of cuts and metrics, Vol. 15 of Algorithms and Combinatorics, Springer, 1997.

    Google Scholar 

  15. Drew, J. H., and Johnson, C. R.: ‘The completely positive and doubly nonnegative completion problems’, Linear Alg. & Its Appl.44 (1998), 85–92.

    MathSciNet  MATH  Google Scholar 

  16. Dym, H., and Gohberg, I.: ‘Extensions of band matrices with band inverses’, Linear Alg. & Its Appl.36 (1981), 1–24.

    MathSciNet  MATH  Google Scholar 

  17. Ellis, R. L., Lay, D. C., and Gohberg, I.: ‘On negative eigenvalues of selfadjoint extensions of band matrices’, Linear Alg. & Its Appl.24 (1988), 15–25.

    MathSciNet  MATH  Google Scholar 

  18. Geelen, J.: ‘Maximum rank matrix completion’, Linear Alg. & Its Appl.288 (1999), 211–217.

    MathSciNet  MATH  Google Scholar 

  19. Glunt, W., Hayden, T. L., and Raydan, M.: ‘Molecular conformations from distance matrices’, J. Comput. Chem.14 (1998), 175–190.

    Google Scholar 

  20. Goemans, M. X.: ‘Semidefinite programming in combinatorial optimization’, Math. Program.79 (1997), 143–161.

    MathSciNet  Google Scholar 

  21. Golumbic, M. C.: Algorithmic theory and perfect graphs, Acad. Press, 1980.

    Google Scholar 

  22. Gray, L. J., and Wilson, D. G.: ‘Nonnegative factorization of positive semidefinite nonnegative matrices’, Linear Alg. & Its Appl.31 (1980), 119–127.

    MathSciNet  MATH  Google Scholar 

  23. Grone, R., Johnson, C. R., Sá, E. M., and Wolkowicz, H.: ‘Positive definite completions of partial hermitian matrices’, Linear Alg. & Its Appl.58 (1984), 109–124.

    MATH  Google Scholar 

  24. Grötschel, M., Lovász, L., and Schrijver, A.: Geometric algorithms and combinatorial optimization, Springer 1988.

    Google Scholar 

  25. Hartfiel, D. J., and Loewy, R.: ‘A determinantal version of the Frobenius-König theorem’, Linear Multilinear Algebra16 (1984), 155–165.

    MathSciNet  MATH  Google Scholar 

  26. Havel, T. F.: ‘An evaluation of computational strategies for use in the determination of protein structure from distance constraints obtained by nuclear magnetic resonance’, Program. Biophys. Biophys. Chem.56 (1991), 43–78.

    Google Scholar 

  27. Helmberg, C., Rendl, F., Vanderbei, R. J., and Wolkowicz, H.: ‘An interior-point method for semidefinite programming’, SIAM J. Optim.6 (1996), 342–361.

    MathSciNet  MATH  Google Scholar 

  28. Helton, J. W., Pierce, S., and Rodman, L.: ‘The ranks of extremal positive semidefinite matrices with given sparsity pattern’, SIAM J. Matrix Anal. Appl.10 (1989), 407–423.

    MathSciNet  MATH  Google Scholar 

  29. Hendrickson, B.: ‘The molecule problem: Determining conformation from pairwise distances’, Techn. Report Dept. Computer Sci. Cornell Univ.90–1159 (1990) PhD Thesis.

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  31. Hendrickson, B.: ‘The molecule problem: exploiting structure in global optimization’, SIAM J. Optim.5 (1995), 835–857.

    MathSciNet  MATH  Google Scholar 

  32. Johnson, C. R.: ‘Matrix completion problems: A survey’, in C. R. Johnson (ed.): Matrix Theory and Appl., Vol. 40 of Proc. Symp. Appl. Math., Amer. Math. Soc., 1990 p. 171–198.

    Google Scholar 

  33. Johnson, C. R., Jones, C., and Kroschel, B.: ‘The distance matrix completion problem: cycle completability’, Linear Multilinear Algebra39 (1995), 195–207.

    MathSciNet  MATH  Google Scholar 

  34. Johnson, C. R., Kroschel, B., and Wolkowicz, H.: ‘An interior-point method for approximate positive semidefinite completions’, Comput. Optim. Appl.9 (1998), 175–190.

    MathSciNet  MATH  Google Scholar 

  35. Johnson, C. R., and Rodman, L.: ‘Inertia possibilities for completions of partial Hermitian matrices’, Linear multilinear algebra16 (1984), 179–195.

    MathSciNet  MATH  Google Scholar 

  36. Johnson, C. R., and Rodman, L.: ‘Completion of matrices to contractions’, J. Funct. Anal.69 (1986), 260–267.

    MathSciNet  MATH  Google Scholar 

  37. Johnson, C. R., and Rodman, L.: ‘Chordal inheritance principles and positive definite completions of partial matrices over function rings’, in I. Gohberg (ed.): Contributions to Operator Theory and its Applications, Birkhäuser 1988 p. 107–127.

    Google Scholar 

  38. Johnson, C. R., and Tarazaga, P.: ‘Connections between the real positive semidefinite and distance matrix completion problems’, Linear Alg. & Its Appl.223/4 (1995), 375–391.

    MathSciNet  Google Scholar 

  39. Khachiyan, L., and Porkolab, L.: ‘Computing integral points in convex semi-algebraic sets’, 38th Annual Symp. Foundations Computer Sci., 1997 p. 162–171.

    Google Scholar 

  40. Kogan, N., and Berman, A.: ‘Characterization of completely positive graphs’, Discrete Math.114 (1993), 297–304.

    MathSciNet  MATH  Google Scholar 

  41. Kuntz, I. D., Thomason, J. F., and Oshiro, C. M.: ‘Distance geometry’, Methods in Enzymologie177 (1993), 159–204.

    Google Scholar 

  42. Laman, G.: ‘On graphs and rigidity of plane skeletal structures’, J. Engin. Math.4 (1970), 331–340.

    MathSciNet  MATH  Google Scholar 

  43. Laurent, M.: ‘Cuts, matrix completions and graph rigidity’, Math. Program.79 (1997), 255–283.

    MathSciNet  Google Scholar 

  44. Laurent, M.: ‘The real positive semidefinite completion problem for series-parallel graphs’, Linear Alg. & Its Appl.252 (1997), 347–366.

    MathSciNet  MATH  Google Scholar 

  45. Laurent, M.: ‘A connection between positive semidefinite and Euclidean distance matrix completion problems’, Linear Alg. & Its Appl.273 (1998), 9–22.

    Article  MathSciNet  MATH  Google Scholar 

  46. Laurent, M.: ‘On the order of a graph and its deficiency in chordality’, CWI ReportPNA-R9801 (1998).

    Google Scholar 

  47. Laurent, M.: ‘A tour d’horizon on positive semidefinite and Euclidean distance matrix completion problems’, in P. M. Pardalos and H. Wolkowicz (eds.): Topics in Semidefinite and Interior-Point Methods, Vol. 18 of Fields Inst. Res. Math. Sci. Commun., Amer. Math. Soc., 1998 p. 51–76.

    Google Scholar 

  48. Laurent, M.: ‘Polynomial instances of the positive semidefinite and Euclidean distance matrix completion problems’, SIAM J. Matrix Anal. Appl. (to appear).

    Google Scholar 

  49. de Leeuw, J., and Heiser, W.: ‘Theory of multidimensional scaling’, in P. R. Krishnaiah and L. N. Kanal (eds.): Handbook Statist., Vol. 2, North-Holland, 1982 p. 285–316

    Google Scholar 

  50. Lovász, L., and Plummer, M. D.: Matching theory, Akad. Kiadó, 1986.

    Google Scholar 

  51. Lovász, L., and Yemini, Y.: ‘On generic rigidity in the plane’, SIAM J. Alg. Discrete Meth.3 (1982), 91–98.

    MATH  Google Scholar 

  52. Lundquist, M. E., and Johnson, C. R.: ‘Linearly constrained positive definite completions’, Linear Alg. & Its Appl.150 (1991), 195–207.

    MathSciNet  MATH  Google Scholar 

  53. Moré, J. J., and Wu, Z.: ‘ε-optimal solutions to distance geometry problems via global continuation’, in P. M. Pardalos and D. Shalloway (eds.): DIMACS, Vol. 23, Amer. Math. Soc., 1996, 151–168.

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  55. Murota, K.: ‘Mixed matrices: Irreducibility and decomposition’, in R. A. Brualdi et al. (eds.): Combinatorial and graph-theoretical problems in linear algebra, Vol. 50 of IMA, Springer 1993 p. 39–71.

    Google Scholar 

  56. Nesterov, Y. E., and Nemirovsky, A. S.: Interior point polynomial algorithms in convex programming: Theory and algorithms, SIAM, 1994.

    Google Scholar 

  57. Pardalos, P. M., and Lin, X.: ‘A tabu based pattern search method for the distance geometry problem’, in F. Giannessis et al. (eds.): Math. Program., Kluwer Acad. Publ., 1997.

    Google Scholar 

  58. Paulsen, V. I., Power, S. C., and Smith, R. R.: ‘Schur products and matrix completions’, J. Funct. Anal.85 (1989), 151–178.

    MathSciNet  MATH  Google Scholar 

  59. Porkolab, L., and Khachiyan, L.: ‘On the complexity of semidefinite programs’, J. Global Optim.10 (1997), 351–365.

    MathSciNet  MATH  Google Scholar 

  60. Ramana, M. V.: ‘An exact duality theory for semidefinite programming and its complexity implications’, Math. Program.77 (1997), 129–162.

    MathSciNet  Google Scholar 

  61. Rose, D. J.: ‘Triangulated graphs and the elimination process’, J. Math. Anal. Appl.32 (1970), 597–609.

    MathSciNet  MATH  Google Scholar 

  62. Saxe, J. B.: ‘Embeddability of weighted graphs in k-space is strongly NP-hard’: Proc. 17th Allerton Conf. Communications, Control and Computing, 1979 p. 480–489.

    Google Scholar 

  63. Schoenberg, I. J.: ‘Remarks to M. Fréchet’s article’ sur la définition axiomatique d’une classe d’espaces vectoriels distanciés applicables vectoriellement sur l’espace de Hilbert”, Ann. of Math.36 (1935), 724–732.

    MathSciNet  Google Scholar 

  64. WEB: ‘http://orion.math.uwaterloo.ca:80/~hwolkowi/henry/software/readme.html’.

    Google Scholar 

  65. WEB: ‘http://www.zib.de/helmberg/semidef.html’.

    Google Scholar 

  66. Woerdeman, H. J.: ‘The lower order of lower triangular operators and minimal rank extensions’, Integral Eq. Operator Theory10 (1987), 859–879.

    MathSciNet  MATH  Google Scholar 

  67. Yemini, Y.: ‘Some theoretical aspects of position-location problems’, Proc. 20th Annual Symp. Foundations Computer Sci., 1979 p. 1–8.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CWI, Kruislaan 413, 1098 SJ, Amsterdam, The Netherlands

    Monique Laurent

Authors
  1. Monique Laurent
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dept. Chemical Engin., Princeton Univ., Princeton, NJ, 08544-5263, USA

    Christodoulos A. Floudas

  2. Center for Applied Optim. Dept. Industrial and Systems Engin., Univ. Florida, Gainesville, FL, 32611, USA

    Panos M. Pardalos

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Kluwer Academic Publishers

About this entry

Cite this entry

Laurent, M. (2001). Matrix Completion Problems . In: Floudas, C.A., Pardalos, P.M. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/0-306-48332-7_271

Download citation

  • DOI: https://doi.org/10.1007/0-306-48332-7_271

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-0-7923-6932-5

  • Online ISBN: 978-0-306-48332-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation