An Analytic Center Cutting Plane Method for Semidefinite Feasibility Problems

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 Γ in the cone of symmetric positive semidefinite matrices. Assume that Γ is defined by an oracle, which for any given m ×m symmetric positive semidefinite matrix Ŷ either confirms that Ŷ ∈ Γ or returns a cut, i.e., a symmetric matrix A such that Γ is in the half-space {Y : A • Y ≤ A • Ŷ }. 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 of the problem. All iterates generated by the algorithm are positive definite matrices. The algorithm has a worst case complexity of O∗(m3/ ) on the total number of cuts to be used, where is the maximum radius of a ball contained by Γ.