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Primal-dual interior-point methods for semidefinite programming: Convergence rates, stability and numerical results. (English) Zbl 0911.65047
Primal-dual interior-point pathfollowing methods for semidefinite programming are considered. Several variants are discussed, based on Newton’s method applied to three equations: primal feasibility, dual feasibility, and some form of centering condition. The focus is on three such algorithms, called the $$XZ$$, $$XZ+ZX$$, and $$Q$$ methods. For the $$XZ+ ZX$$ and $$Q$$ algorithms, the Newton system is well defined and its Jacobian is nonsingular at the solution, under nondegeneracy assumptions. The associated Schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption. Practical aspects are discussed, including Mehrotra predictor-corrector variants and issues of numerical stability. 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 rapidly, and achieves higher accuracy.
Reviewer: J.Guddat (Berlin)

MSC:
 65K05 Numerical mathematical programming methods 90C25 Convex programming
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