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Positive Semidefinite Matrices

  • Abdo Y. Alfakih
Chapter

Abstract

Positive semidefinite (PSD) and positive definite (PD) matrices are closely connected with Euclidean distance matrices. Accordingly, they play a central role in this monograph. This chapter reviews some of the basic results concerning these matrices. Among the topics discussed are various characterizations of PSD and PD matrices, theorems of the alternative for the semidefinite cone, the facial structures of the semidefinite cone and spectrahedra, as well as the Borwein–Wolkowicz facial reduction scheme.

References

  1. 14.
    A.Y. Alfakih, On Farkas lemma and dimensional rigidity of bar frameworks. Linear Algebra Appl. 486, 504–522 (2015)MathSciNetCrossRefGoogle Scholar
  2. 26.
    F. Alizadeh, J.A. Haeberly, M.L. Overton, Complementarity and nondegeneracy in semidefinite programming. Math. Program. Ser. B 77, 111–128 (1997)MathSciNetzbMATHGoogle Scholar
  3. 34.
    G. Barker, Theory of cones. Linear Algebra Appl. 39, 263–291 (1981)MathSciNetCrossRefGoogle Scholar
  4. 35.
    G.P. Barker, D. Carlson, Cones of diagonally dominant matrices. Pac. J. Math. 57, 15–31 (1975)MathSciNetCrossRefGoogle Scholar
  5. 36.
    G. Barker, R. Hill, R. Haertel, On the completely positive and positive semidefinite preserving cones. Linear Algebra Appl. 56, 221–229 (1984)MathSciNetCrossRefGoogle Scholar
  6. 49.
    J.M. Borwein, H. Wolkowicz, Facial reduction for a non-convex programming problem. J. Austral. Math. Soc. Ser. A 30, 369–380 (1981)CrossRefGoogle Scholar
  7. 50.
    J.M. Borwein, H. Wolkowicz, Regularizing the abstract convex program. J. Math. Anal. Appl. 83, 495–530 (1981)MathSciNetCrossRefGoogle Scholar
  8. 54.
    Y.-L. Cheung, Preprocessing and Reduction for Semidefinite Programming via Facial Reduction: Theory and Practice. PhD thesis, University of Waterloo, 2013Google Scholar
  9. 74.
    D. Drusvyatskiy, H. Wolkowicz, The many faces of degeneracy in conic optimization. Found. Trends Optim. 3, 77–170 (2017)CrossRefGoogle Scholar
  10. 75.
    D. Drusvyatskiy, G. Pataki, H. Wolkowicz, Coordinate shadows of semidefinite and Euclidean distance matrices. SIAM J. Optim. 25, 1160–1178 (2015)MathSciNetCrossRefGoogle Scholar
  11. 76.
    M. Dur, B. Jargalsaikhan, G. Still, Genericity results in linear conic programming, a tour d’horizon. Math. Oper. Res. 42, 77–94 (2017)MathSciNetCrossRefGoogle Scholar
  12. 108.
    R. Hill, S. Waters, On the cone of positive semidefinite matrices. Linear Algebra Appl. 90, 81–88 (1987)MathSciNetCrossRefGoogle Scholar
  13. 136.
    M. Liu, G. Pataki, Exact duality in semidefinite programming based on elementary reformations. SIAM J. Optim. 25, 1441–1454 (2015)MathSciNetCrossRefGoogle Scholar
  14. 137.
    L. Lovász, Semidefinite programs and combinatorial optimization, in Recent Advances in Algorithms and Combinatorics (Springer, New York, 2003), pp. 137–194zbMATHGoogle Scholar
  15. 154.
    G. Pataki, The geometry of semidefinite programing, in Handbook of Semidefinite Programming: Theory, Algorithms and Applications, ed. by H. Wolkowicz, R. Saigal, L. Vandenberghe (Kluwer Academic Publishers, Boston, 2000), pp. 29–65CrossRefGoogle Scholar
  16. 155.
    J.E. Prussing, The principal minor test for semidefinite matrices. J. Guid. Control Dyn. 9, 121–122 (1986)CrossRefGoogle Scholar
  17. 156.
    M. Ramana, A.J. Goldman, Some geometric results in semi-definite programming. J. Glob. Optim. 7, 33–50 (1995)CrossRefGoogle Scholar
  18. 165.
    H. Schneider, Positive operators and an inertia theorem. Numer. Math. 7, 11–17 (1965)MathSciNetCrossRefGoogle Scholar
  19. 181.
    J.F. Sturm, Error bounds for linear matrix inequalities. SIAM J. Optim. 10, 1228–1248 (2000)MathSciNetCrossRefGoogle Scholar
  20. 198.
    H. Wolkowicz, R. Saigal, L. Vandenberghe (eds.), Handbook of Semidefinite Programming. Theory, Algorithms and Applications (Kluwer Academic Publishers, Boston, 2000)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Abdo Y. Alfakih
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of WindsorWindsorCanada

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