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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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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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