An Analytic Center Cutting Plane Method for Semideenite Feasibility Problems

Semideenite feasibility problems arise in many areas of operations research. The abstract form of these problems can be described as nding a point in a nonempty bounded convex body ? in the cone of symmetric positive semideenite matrices. Assume that ? is deened by an oracle, which for any given m m symmetric positive semideenite matrix ^ Y either connrms that ^ Y 2 ? or returns a cut, i.e., a symmetric matrix A such that ? is in the half-space fY : A Y A ^ Y g: 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 deened 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 nds a solution of the problem. All iterates generated by the algorithm are positive deenite matrices. The algorithm has a worst case complexity of O (m 3 == 2) on the total number of cuts to be used, where is the maximum radius of a ball contained by ?.