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Complementarity and nondegeneracy in semidefinite programming

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Abstract

Primal and dual nondegeneracy conditions are defined for semidefinite programming. Given the existence of primal and dual solutions, it is shown that primal nondegeneracy implies a unique dual solution and that dual nondegeneracy implies a unique primal solution. The converses hold if strict complementarity is assumed. Primal and dual nondegeneracy assumptions do not imply strict complementarity, as they do in LP. The primal and dual nondegeneracy assumptions imply a range of possible ranks for primal and dual solutionsX andZ. This is in contrast with LP where nondegeneracy assumptions exactly determine the number of variables which are zero. It is shown that primal and dual nondegeneracy and strict complementarity all hold generically. Numerical experiments suggest probability distributions for the ranks ofX andZ which are consistent with the nondegeneracy conditions.

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References

  1. F. Alizadeh, J.-P.A. Haeberly and M.L. Overton, Primal-dual interior-point methods for semidefinite programming: Convergence rates, stability and numerical results,SIAM Journal on Optimization, to appear.

  2. E. Anderson and P. Nash,Linear Programming in Infinite Dimensional Spaces (John Wiley, New York, 1987).

    MATH  Google Scholar 

  3. V.I. Arnold, On matrices depending on parameters,Russian Mathematical Surveys 26 (1971) 29–43.

    Article  Google Scholar 

  4. W.M. Boothby,An Introduction to Differentiable Manifolds and Riemannian Geometry (Academic Press, New York, 1975).

    MATH  Google Scholar 

  5. V. Guillemin and A. Pollack,Differential Topology (Prentice-Hall, Englewood Cliffs, NJ, 1974).

    MATH  Google Scholar 

  6. D. Husemoller,Fibre Bundles (Springer, New York, 2nd ed., 1975).

    MATH  Google Scholar 

  7. Y. Nesterov and A. Nemirovskii,Interior Point Polynomial Algorithms in Convex Programming (SIAM, Philadelphia, PA, 1994).

    MATH  Google Scholar 

  8. M.L. Overton and R.S. Womersley, Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices,Mathematical Programming 62 (1993) 321–357.

    Article  MathSciNet  Google Scholar 

  9. M.L. Overton and R.S. Womersley, Second derivatives for optimizing eigenvalues of symmetric matrices,SIAM Journal on Matrix Analysis and Applications 16 (1995) 697–718.

    Article  MATH  MathSciNet  Google Scholar 

  10. G. Pataki, On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues, Tech. Report MSRR-604 (revised), Graduate School of Industrial Administration, Carnegie-Mellon University, 1995.

  11. A. Shapiro and M.K.H. Fan, On eigenvalue optimization,SIAM Journal on Optimization 3 (1995) 552–568.

    Article  MathSciNet  Google Scholar 

  12. A. Shapiro, First and second order analysis of nonlinear semidefinite programs,Mathematical Programming 77 (1997) 301–320 (this issue).

    Article  MathSciNet  Google Scholar 

  13. O. Taussky, Positive-definite matrices and their role in the study of the characteristic roots of general matrices,Advances in Mathematics 2 (1967) 175–186.

    Article  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. RUTCOR, Rutgers University, New Brunswick, NJ, USA

    Farid Alizadeh

  2. Mathematics Department, Fordham University, Bronx, NY, USA

    Jean-Pierre A. Haeberly

  3. Computer Science Department, Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., 10012, New York, NY, USA

    Michael L. Overton

Authors
  1. Farid Alizadeh
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  2. Jean-Pierre A. Haeberly
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  3. Michael L. Overton
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Corresponding author

Correspondence to Michael L. Overton.

Additional information

Supported in part by the U.S. National Science Foundation grant CCR-9625955.

Supported in part by U.S. National Science Foundation grant CCR-9501941 and the U.S. Office of Naval Research grant N00014-96-1-0704.

Supported in part by U.S. National Science Foundation grant CCR-9401119.

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Alizadeh, F., Haeberly, JP.A. & Overton, M.L. Complementarity and nondegeneracy in semidefinite programming. Mathematical Programming 77, 111–128 (1997). https://doi.org/10.1007/BF02614432

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