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Complementarity and Nondegeneracy in Semidefinite Programming \Lambda

Summary: Complementarity and Nondegeneracy in
Semidefinite Programming \Lambda
Farid Alizadeh y Jean­Pierre A. Haeberly z Michael L. Overton x
March 1995
Submitted to Math. Programming
Primal and dual nondegeneracy conditions are defined for semidef­
inite 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 im­
ply 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 nondegeneracy


Source: Alizadeh, Farid - Rutgers Center for Operations Research, Rutgers University


Collections: Mathematics