Semidefinite feasibility problems arise in many areas of operations research. The abstract form of these problems can be described as finding a point in a nonempty bounded convex body Gamma in the cone of symmetric positive semidefinite matrices. Assume that Gamma is defined by an oracle, which for any given m x m symmetric positive semidefinite matrix (Y) over cap either confirms that (Y) over cap epsilon Gamma or returns a cut, i.e., a symmetric matrix A such that Gamma is in the half-space {Y : A . Y less than or equal to A . (Y) over cap}. We study an analytic center cutting plane algorithm for this problem. At each iteration, the algorithm computes an approximate analytic center of a working set defined by the cutting plane system generated in the previous iterations. If this approximate analytic center is a solution, then the algorithm terminates; otherwise the new cutting plane returned by the oracle is added into the system. As the number of iterations increases, the working set shrinks and the algorithm eventually finds a solution to the problem. All iterates generated by the algorithm are positive definite matrices. The algorithm has a worst-case complexity of O*(m(3)/epsilon(2)) on the total number of cuts to be used, where epsilon is the maximum radius of a ball contained by Gamma. [References: 10]
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