Skip to main content
SpringerLink
Log in

Matrix Completion Problems

  • Reference work entry
Encyclopedia of Optimization

Article Outline

Keywords

Introduction

Positive Semidefinite Completion Problem

Euclidean Distance Matrix Completion Problem

Completion to Completely Positive and Contraction Matrices

Rank Completions

See also

References

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 2,500.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 2,499.99
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

Institutional subscriptions

References

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Barahona F, Mahjoub AR (1986) On the cut polytope. Math Program 36:157–173

    Article  MathSciNet  MATH  Google Scholar 

  6. Barrett WW, Johnson CR, Loewy R (1996) The real positive definite completion problem: cycle completability. Memoirs Amer Math Soc, vol 584. Amer. Math. Soc., Providence, RI

    Google Scholar 

  7. Barrett WW, Johnson CR, Lundquist M (1989) Determinantal formulas for matrix completions associated with chordal graphs. Linear Alg & Its Appl 121:265–289

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Barvinok AI (1993) Feasibility testing for systems of real quadratic equations. Discrete Comput Geom 10:1–13

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Blumenthal LM (1953) Theory and applications of distance geometry. Oxford Univ. Press, Oxford

    MATH  Google Scholar 

  12. Cohen N, Johnson CR, Rodman L, Woederman HJ (1989) Ranks of completions of partial matrices. In: Dym H, et al (eds) The Gohberg Anniv. Coll. vol I, Birkhäuser, Basel, pp 165–185

    Google Scholar 

  13. Crippen GM, Havel TF (1988) Distance geometry and molecular conformation. Res. Studies Press, Taunton, UK

    MATH  Google Scholar 

  14. Deza M, Laurent M (1997) Geometry of cuts and metrics. Algorithms and Combinatorics, vol 15. Springer, Berlin

    MATH  Google Scholar 

  15. Drew JH, Johnson CR (1998) The completely positive and doubly nonnegative completion problems. Linear Alg & Its Appl 44:85–92

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Ellis RL, Lay DC, Gohberg I (1988) On negative eigenvalues of selfadjoint extensions of band matrices. Linear Alg & Its Appl 24:15–25

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Glunt W, Hayden TL, Raydan M (1998) Molecular conformations from distance matrices. J Comput Chem 14:175–190

    Google Scholar 

  20. Goemans MX (1997) Semidefinite programming in combinatorial optimization. Math Program 79:143–161

    MathSciNet  Google Scholar 

  21. Golumbic MC (1980) Algorithmic theory and perfect graphs. Acad. Press, New York

    MATH  Google Scholar 

  22. Gray LJ, Wilson DG (1980) Nonnegative factorization of positive semidefinite nonnegative matrices. Linear Alg & Its Appl 31:119–127

    Article  MathSciNet  MATH  Google Scholar 

  23. Grone R, Johnson CR, Sá EM, Wolkowicz H (1984) Positive definite completions of partial hermitian matrices. Linear Alg & Its Appl 58:109–124

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  25. Hartfiel DJ, Loewy R (1984) A determinantal version of the Frobenius-König theorem. Linear Multilinear Algebra 16:155–165

    Article  MathSciNet  MATH  Google Scholar 

  26. Havel TF (1991) 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 56:43–78

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. Johnson CR (1990) Matrix completion problems: A survey. In: Johnson CR (ed) Matrix Theory and Appl. Proc Symp Appl Math. Amer. Math. Soc., Providence, RI, pp 171–198

    Google Scholar 

  33. Johnson CR, Jones C, Kroschel B (1995) The distance matrix completion problem: cycle completability. Linear Multilinear Algebra 39:195–207

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Johnson CR, Rodman L (1984) Inertia possibilities for completions of partial Hermitian matrices. Linear multilinear algebra 16:179–195

    Article  MathSciNet  MATH  Google Scholar 

  36. Johnson CR, Rodman L (1986) Completion of matrices to contractions. J Funct Anal 69:260–267

    Article  MathSciNet  MATH  Google Scholar 

  37. Johnson CR, Rodman L (1988) Chordal inheritance principles and positive definite completions of partial matrices over function rings. In: Gohberg I (ed) Contributions to Operator Theory and its Applications. Birkhäuser, Basel, pp 107–127

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  39. Khachiyan L, Porkolab L (1997) Computing integral points in convex semi-algebraic sets. 38th Annual Symp. Foundations Computer Sci., pp 162–171

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  41. Kuntz ID, Thomason JF, Oshiro CM (1993) Distance geometry. Methods in Enzymologie 177:159–204

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  46. Laurent M (1998) On the order of a graph and its deficiency in chordality. CWI Report PNA-R9801

    Google Scholar 

  47. Laurent M (1998) A tour d'horizon on positive semidefinite and Euclidean distance matrix completion problems. In: Pardalos PM, Wolkowicz H (eds) Topics in Semidefinite and Interior-Point Methods. Fields Inst Res Math Sci Commun. Amer. Math. Soc., Providence, RI, pp 51–76

    Google Scholar 

  48. Laurent M (2000) Polynomial instances of the positive semidefinite and Euclidean distance matrix completion problems. SIAM J Matrix Anal Appl 22(3)(874–894

    Article  MathSciNet  MATH  Google Scholar 

  49. de Leeuw J, Heiser W (1982) Theory of multidimensional scaling. In: Krishnaiah PR, Kanal LN (eds) Handbook Statist., vol 2, North-Holland, Amsterdam, pp 285–316

    Google Scholar 

  50. Lovász L, Plummer MD (1986) Matching theory. Akad. Kiadó, Budapest

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  52. Lundquist ME, Johnson CR (1991) Linearly constrained positive definite completions. Linear Alg & Its Appl 150:195–207

    Article  MathSciNet  MATH  Google Scholar 

  53. Moré JJ, Wu Z (1996) ϵ-optimal solutions to distance geometry problems via global continuation. In: Pardalos PM, Shalloway D (eds) DIMACS, vol 23, Amer. Math. Soc., Providence, RI, 151–168

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  55. Murota K (1993) Mixed matrices: Irreducibility and decomposition. In: Brualdi RA et al (eds) Combinatorial and graph-theoretical problems in linear algebra. IMA. Springer, Berlin, pp 39–71

    Google Scholar 

  56. Nesterov YE, Nemirovsky AS (1994) Interior point polynomial algorithms in convex programming: Theory and algorithms. SIAM, Philadelphia

    MATH  Google Scholar 

  57. Pardalos PM, Lin X (1997) A tabu based pattern search method for the distance geometry problem. In: Giannessis F et al (eds) Math. Program. Kluwer, Dordrecht

    Google Scholar 

  58. Paulsen VI, Power SC, Smith RR (1989) Schur products and matrix completions. J Funct Anal 85:151–178

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  62. Saxe JB (1979) Embeddability of weighted graphs in k-space is strongly NP-hard. In: Proc. 17th Allerton Conf. Communications, Control and Computing, pp 480–489

    Google Scholar 

  63. Schoenberg IJ (1935) 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 36:724–732

    Article  MathSciNet  Google Scholar 

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

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

  66. Woerdeman HJ (1987) The lower order of lower triangular operators and minimal rank extensions. Integral Eq Operator Theory 10:859–879

    Article  MathSciNet  MATH  Google Scholar 

  67. Yemini Y (1979) Some theoretical aspects of position-location problems. Proc. 20th Annual Symp. Foundations Computer Sci., pp 1–8

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CWI, 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. Department of Chemical Engineering, Princeton University, Princeton, NJ, 08544-5263, USA

    Christodoulos A. Floudas

  2. Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, 32611-6595, USA

    Panos M. Pardalos

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag

About this entry

Cite this entry

Laurent, M. (2008). Matrix Completion Problems . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_355

Download citation

Publish with us

Policies and ethics

Search

Navigation