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Journal of Global Optimization

, Volume 7, Issue 1, pp 33–50 | Cite as

Some geometric results in semidefinite programming

  • Motakuri Ramana
  • A. J. Goldman
Article

Abstract

The purpose of this paper is to develop certain geometric results concerning the feasible regions of Semidefinite Programs, called hereSpectrahedra.

We first develop a characterization for the faces of spectrahedra. More specifically, given a pointx in a spectrahedron, we derive an expression for the minimal face containingx. Among other things, this is shown to yield characterizations for extreme points and extreme rays of spectrahedra. We then introduce the notion of an algebraic polar of a spectrahedron, and present its relation to the usual geometric polar.

Key words

Semidefinite programming convex geometry 

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References

  1. 1.
    F. Alizadeh (1991), Combinatorial Optimization with Interior Point Methods and Semi-Definite Matrices, Ph.D. Thesis, Computer Science Department, University of Minnesota, Minneapolis, Minnesota, 1991.Google Scholar
  2. 2.
    F. Alizadeh (1995), Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization,SIAM J. Optimization 5, No. 1.Google Scholar
  3. 3.
    G. P. Barker and D. Carlson (1975), Cones of Diagonally Dominant Matrices,Pac. J. Math. 57, 15–31.Google Scholar
  4. 4.
    P. Binding (1990), Simultaneous Diagonalization of Several Hermitian Matrices,SIAM J. Matrix Anal Appl. 11, 531–536.Google Scholar
  5. 5.
    P. Binding and C.-K. Li (1991), Joint Ranges of Hermitian Matrices and Simultaneous Diagonalization,Linear Algebra Appl. 151, 157–167.Google Scholar
  6. 6.
    A. Ben-Israel, A. Charnes, and K. Kortanek, (1969) Duality and Asymptotic Solvability over Cones,Bull. of AMS 75, 318–324.Google Scholar
  7. 7.
    A. Berman (1973),Cone, Matrices, and Mathematical Programming; Lecture Notes in Economics and Mathematical Systems, Springer.Google Scholar
  8. 8.
    J. Borwein and H. Wolkowicz (1981), Characterization of Optimality for the Abstract Convex Program with Finite Dimensional Range,J. Austral. Math. Soc., Series A30, 390–411.Google Scholar
  9. 9.
    J. Cullum, W. E. Donath, and P. Wolfe (1975), The Minimization of Certain Nondifferentiable Sums of Eigenvalue Problems,Math. Prog. Study 3, 35–55.Google Scholar
  10. 10.
    C. Delorme and S. Poljak (1993), Combinatorial Properties and the Complexity of a Max-Cut Approximation,Europ. J. Combinatorics 14, 313–333.Google Scholar
  11. 11.
    R. Fletcher (1985), Semi-Definite Matrix Constraints in Optimization,SIAM J. Control and Optimization 23, 493–513.Google Scholar
  12. 12.
    M. X. Goemans and D. P. Williamson (1995), Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming,Submitted to J. ACM. (contact goemans@math.mit.edu for copies)Google Scholar
  13. 13.
    R. Grone, S. Pierce, and W. Watkins (1990), Extremal Correlation Matrices,Linear Algebra and its Applications 134, pp. 63–70.Google Scholar
  14. 14.
    M. Grötschel, L. Lovásza, and A. Schrijver (1984), Polynomial Algorithms for Perfect Graphs,Annals of Discrete Mathematics 21, C. Berge and V. Chvátal, eds., North Holland.Google Scholar
  15. 15.
    M. Grötschel, L. Lovász, and A. Schrijver (1988),Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, Berlin.Google Scholar
  16. 16.
    R. Horn and C.R. Johnson,Matrix Analysis, Cambridge University Press, Cambridge, 1985.Google Scholar
  17. 17.
    M. Kojima, S. Kojima, and S. Hara, Linear Algebra for Semidefinite Programming, TR B-290, Research Reports on Information Sciences, Tokyo Institute of Technology, Tokyo, Japan, 1994.Google Scholar
  18. 18.
    M. Laurent and S. Poljak, On a Positive Semideflnite Relaxation of the Cut Polytope, Technical Report, LIENS-93-27, Ecole Normale Supérieure, France, 1993. (Contact monique@cwi.nl for copies)Google Scholar
  19. 19.
    L. Lovász and A. Schrijver, Cones of Matrices and Setfunctions, and 0–1 Optimization,SIAM J. Optimization 1 (1991).Google Scholar
  20. 20.
    Y. Nesterov and A. Nemirovskii,Interior Point Polynomial Methods for Convex Programming: Theory and Applications, SIAM, 1994.Google Scholar
  21. 21.
    M. L. Overton, Large-Scale Optimization of Eigenvalues,SIAM J. Optimization 2 (1992), pp. 88–120.Google Scholar
  22. 22.
    M. L. Overton and R. S. Womersley, Optimality Conditions and Duality Theory for Minimizing Sums of the Largest Eigenvalues of Symmetric Matrices,Math. Prog., Series B62 (1993), pp. 321–357.Google Scholar
  23. 23.
    P. M. Pardalos and S. A. Vavasis (1992), Open Questions in Complexity Theory for Numerical Optimization,Math. Prog. 57(2), 337–339.Google Scholar
  24. 24.
    G. Pataki, Algorithms for Linear Programs over Cones and Semidefinite Programming, Technical Report, GSIA, Carnegie-Mellon University, Pittsburgh, 1993. (contact gabor@magrathea.gsia.cmu.edu for copies)Google Scholar
  25. 25.
    G. Pataki, On the Facial Structure of Cone-LP's and Semidefinite Programs, Management Science Research Report # MSRR-595, GSIA, Carnegie-Mellon University, Pittsburgh, 1994.Google Scholar
  26. 26.
    M. Ramana (1995), An Exact Duality Theory for Semidefinite Programming and its Complexity Implications, DIMACS TR 95-02R (http://www.dimacs.edu), Rutgers University; Submitted toMath Programming.Google Scholar
  27. 27.
    M. V. Ramana (1993), An algorithmic analysis of multiquadratic and Semidefinite programming problems, Ph.D. Thesis, The Johns Hopkins University, Baltimore, 1993.Google Scholar
  28. 28.
    M. V. Ramana and A. J. Goldman, Cutting Plane Techniques for Multiquadratic Programming, Under Preparation.Google Scholar
  29. 29.
    M. V. Ramana and A. J. Goldman, Quadratic Maps with Convex Images, Submitted to Math of OR.Google Scholar
  30. 30.
    T. R. Rockafellar,Convex Analysis, Princeton University Press, Princeton, 1970.Google Scholar
  31. 31.
    L. Vandenberghe and S. Boyd (1994), Positive-Definite Programming,Mathematical Programming: State of the Art 1994, J. R. Birge and K. G. Murty (eds.), U. of Michigan.Google Scholar
  32. 32.
    H. Wolkowicz, Some Applications of Optimization in Matrix Theory,Linear Algebra and its Applications 40 (1981), 101–118.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Motakuri Ramana
    • 1
  • A. J. Goldman
    • 2
  1. 1.Center for Operations Research (RUTCOR)Rutgers UniversityNew BrunswickUSA
  2. 2.Mathematical Sciences DepartmentThe Johns Hopkins UniversityBaltimoreUSA

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