In this chapter we study interior-point primal-dual path-following algorithms for solving the semidefinite programming (SDP) problem. In contrast to linear programming, there are several ways one can define the Newton-type search directions used by these algorithms. We discuss several ways in which this can done by motivating and introducing several search directions and families of directions. Polynomial convergence results for short- and long-step path-following algorithms using the Monteiro and Zhang family of directions are derived in detail; similar results are only surveyed, without proofs, for the Kojima, Shindoh and Hara family and the Monteiro and Tsuchiya family.
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