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

S. K. Chua, K. C. Toh and G. Y. Zhao
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S. K. Chua: National University of Singapore
K. C. Toh: National University of Singapore
G. Y. Zhao: National University of Singapore

Journal of Optimization Theory and Applications, 2004, vol. 123, issue 2, No 4, 318 pages

Abstract: 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 {Y∈S m : A●Y≤ A●YŶ - ξ} ⊃ Γ, where ξ ≥ 0. If 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 ∈ > 0, the algorithm obtains eventually a point in Γ, with a worst-case complexity of O *(m 3/∈2) on the total number of cuts generated.

Keywords: Analytic centers; cutting-plane methods; semidefinite feasibility problems; deep cuts (search for similar items in EconPapers)
Date: 2004
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DOI: 10.1007/s10957-004-5150-4

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