Analytic center cutting-plane method with deep cuts for semidefinite feasibility problems

Abstract

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 Γ 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 Ŷ that is an approximate analytic center of the current working set is chosen. We assume that there exists an oracle which either confirms that Ŷ ∈ Γ or returns a cut A ∈ S m such that {Y ∈ S m: A• Y ≤ A• Ŷ - ξ} ⊃Γ, where ξ ≥ 0. Ŷ ∉ Γ, 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 Γ contains a ball with radius ε > 0, the algorithm obtains eventually a point in Γ, with a worst-case complexity of O*(m 3/ε 2) on the total number of cuts generated.