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PrimalDual InteriorPoint Methods for Semidefinite Programming: Convergence Rates, Stability and

Summary: Primal­Dual Interior­Point Methods for Semidefinite
Programming: Convergence Rates, Stability and
Numerical Results \Lambda
Farid Alizadeh y Jean­Pierre A. Haeberly z Michael L. Overton x
May 16, 1996
Primal­dual interior­point path­following methods for semidefinite
programming (SDP) are considered. Several variants are discussed,
based on Newton's method applied to three equations: primal feasibil­
ity, dual feasibility, and some form of centering condition. The focus is
on three such algorithms, called respectively the XZ, XZ + ZX and
Q methods. For the XZ + ZX and Q algorithms, the Newton system
is well­defined and its Jabobian is nonsingular at the solution, under
nondegeneracy assumptions. The associated Schur complement matrix
has an unbounded condition number on the central path, under the non­
degeneracy assumptions and an additional rank assumption. Practical
aspects are discussed, including Mehrotra predictor­corrector variants
and issues of numerical stability. We observe that, compared to the
other methods considered, the XZ + ZX method is more robust with
respect to its ability to step close to the boundary, converges more


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


Collections: Mathematics