Analytic Center Cutting-Plane Method with Deep Cuts for Semidefinite Feasibility Problems

An analytic center cutting-plane method with deep cuts for semidefinite feasibility problems is presented. Our objective in these problems is to find a point in a nonempty bounded convex set in the cone of symmetric positive-semidefinite matrices. The cutting plane method achieves this by the following iterative scheme. At each iteration, a query point that is an approximate analytic center of the current working set is chosen. We assume that there exists an oracle which either confirms that or returns a cut A S ^m {YS ^m : AY AY - } , where 0. If , an approximate analytic center of the new working set, defined by adding the new cut to the preceding working set, is then computed via a primal Newton procedure. Assuming that contains a ball with radius > 0, the algorithm obtains eventually a point in , with a worst-case complexity of O ^*(m ³/²) on the total number of cuts generated.