A Multiple-Cut Analytic Center Cutting Plane Method for Semidefinite Feasibility Problems

We consider the problem of finding a point in a nonempty bounded convex body Γ in the cone of symmetric positive semidefinite matrices S + . Assume that Γ is defined by a separating oracle, which, for any given m×m symmetric matrix Ŷ , either confirms that Ŷ ∈ Γ or returns several selected cuts, i.e., a number of symmetric matrices Ai, i = 1, ..., p, p ≤ pmax, such that Γ is in the polyhedron {Y ∈ S + | Ai • Y ≤ Ai • Ŷ , i = 1, ..., p}. We present a multiple-cut analytic center cutting plane algorithm. Starting from a trivial initial point, the algorithm generates a sequence of positive definite matrices which are approximate analytic centers of a shrinking polytope in S + . The algorithm terminates with a point in Γ within O(mpmax/ ) Newton steps (to leading order), where is the maximum radius of a ball contained in Γ.