### 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 solutions X and Z. 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 of X and Z which are consistent with the no(c)eracy conditions.

Original language | English (US) |
---|---|

Pages (from-to) | 111-128 |

Number of pages | 18 |

Journal | Mathematical Programming |

Volume | 77 |

Issue number | 2 |

State | Published - May 1 1997 |

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### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Software
- Management Science and Operations Research
- Safety, Risk, Reliability and Quality
- Mathematics(all)
- Applied Mathematics

### Cite this

*Mathematical Programming*,

*77*(2), 111-128.

**Complementarity and nondegeneracy in semidefinite programming.** / Alizadeh, Farid; Haeberly, Jean Pierre A; Overton, Michael L.

Research output: Contribution to journal › Article

*Mathematical Programming*, vol. 77, no. 2, pp. 111-128.

}

TY - JOUR

T1 - Complementarity and nondegeneracy in semidefinite programming

AU - Alizadeh, Farid

AU - Haeberly, Jean Pierre A

AU - Overton, Michael L.

PY - 1997/5/1

Y1 - 1997/5/1

N2 - 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 solutions X and Z. 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 of X and Z which are consistent with the no(c)eracy conditions.

AB - 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 solutions X and Z. 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 of X and Z which are consistent with the no(c)eracy conditions.

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UR - http://www.scopus.com/inward/citedby.url?scp=0031141136&partnerID=8YFLogxK

M3 - Article

VL - 77

SP - 111

EP - 128

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 2

ER -