Complementarity and nondegeneracy in semidefinite programming

Farid Alizadeh, Jean Pierre A Haeberly, Michael L. Overton

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)111-128
Number of pages18
JournalMathematical Programming
Volume77
Issue number2
StatePublished - May 1 1997

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Nondegeneracy
Semidefinite Programming
Complementarity
Strict Complementarity
Dual Solutions
Imply
Probability distributions
Semidefinite programming
Converse
Probability Distribution
Numerical Experiment
Experiments
Zero
Range of data

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

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

In: Mathematical Programming, Vol. 77, No. 2, 01.05.1997, p. 111-128.

Research output: Contribution to journalArticle

Alizadeh, F, Haeberly, JPA & Overton, ML 1997, 'Complementarity and nondegeneracy in semidefinite programming', Mathematical Programming, vol. 77, no. 2, pp. 111-128.
Alizadeh, Farid ; Haeberly, Jean Pierre A ; Overton, Michael L. / Complementarity and nondegeneracy in semidefinite programming. In: Mathematical Programming. 1997 ; Vol. 77, No. 2. pp. 111-128.
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