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We study the semide nite programming problem (SDP), i.e the optimization problem of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semide nite. First we review the classical cone duality as is specialized to SDP. Next we present an interior point algorithm which converges to the optimal solution in polynomial time. The approach is a direct extension of Ye's projective method for linear programming. We also argue that many known interior point methods for linear programs can be transformed in a mechanical way to algorithms for SDP with proofs of convergence and polynomial time complexity also carrying over in a similar fashion. Finally we study the signi cance of these results in a variety of combinatorialoptimizationproblems including the general 0-1 integer programs, the maximum clique and maximum stable set problems in perfect graphs, the maximum k-partite subgraph problem in graphs, and various graph partitioning and cut problems. As a result, we present barrier oracles for certain combinatorial optimization problems (in particular, clique and stable set problem for perfect graphs) whose linear programming formulation requires exponentially many inequalities. Existence of such barrier oracles refutes the commonly believed notion that in order to solve a combinatorial optimization problem with interior point methods, one needs its linear programming formulation explicitly.

Primal-dual interior-point path-following methods for semide nite programming (SDP) 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 respectively the XZ, XZ+ZX and Q methods. For the XZ+ZX and Q algorithms, the Newton system is well-de ned 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.

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