Interior-point methods in semidefinite programming (SDP) require the solution of a sequence of linear systems which are used to derive the search directions. Safeguards are typically required in order to handle rank-deficient Jacobians and free variables. We generalize the primal-dual regularization of \cite{friedlander-orban-2012} to SDP and show that it is possible to recover an optimal solution of the original primal-dual pair via inaccurate solves of a sequence of regularized SDPs for both the NT and dual HKM directions. Computationally, a sparse $LDL^T$ factorization may be used on a sparse augmented system instead of the more costly symmetric indefinite factorization. Benefits of our approach include increased robustness and a simpler implementation. Our method does not require the constraints to be linearly independent and does not assume that Slater's condition holds. We report numerical experience on standard problems that illustrate our findings.
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Cahier du GERAD G-2012-12, GERAD, Montreal, QC., Canada
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