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

Author

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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)

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.

Suggested Citation

  • S. K. Chua & K. C. Toh & G. Y. Zhao, 2004. "Analytic Center Cutting-Plane Method with Deep Cuts for Semidefinite Feasibility Problems," Journal of Optimization Theory and Applications, Springer, vol. 123(2), pages 291-318, November.
    • Handle: RePEc:spr:joptap:v:123:y:2004:i:2:d:10.1007_s10957-004-5150-4
    DOI: 10.1007/s10957-004-5150-4
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    References listed on IDEAS

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    1. Jie Sun & Kim-Chuan Toh & Gongyun Zhao, 2002. "An Analytic Center Cutting Plane Method for Semidefinite Feasibility Problems," Mathematics of Operations Research, INFORMS, vol. 27(2), pages 332-346, May.
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    Cited by:

    1. Mohammad R. Oskoorouchi & Hamid R. Ghaffari & Tamás Terlaky & Dionne M. Aleman, 2011. "An Interior Point Constraint Generation Algorithm for Semi-Infinite Optimization with Health-Care Application," Operations Research, INFORMS, vol. 59(5), pages 1184-1197, October.
    2. Chee-Khian Sim, 2019. "Interior point method on semi-definite linear complementarity problems using the Nesterov–Todd (NT) search direction: polynomial complexity and local convergence," Computational Optimization and Applications, Springer, vol. 74(2), pages 583-621, November.
    3. Mohammad R. Oskoorouchi & Jean-Louis Goffin, 2005. "An Interior Point Cutting Plane Method for the Convex Feasibility Problem with Second-Order Cone Inequalities," Mathematics of Operations Research, INFORMS, vol. 30(1), pages 127-149, February.
    4. John Mitchell & Vasile Basescu, 2008. "Selective Gram–Schmidt orthonormalization for conic cutting surface algorithms," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 67(1), pages 91-115, February.
    5. Vasile L. Basescu & John E. Mitchell, 2008. "An Analytic Center Cutting Plane Approach for Conic Programming," Mathematics of Operations Research, INFORMS, vol. 33(3), pages 529-551, August.

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    1. Chee-Khian Sim, 2019. "Interior point method on semi-definite linear complementarity problems using the Nesterov–Todd (NT) search direction: polynomial complexity and local convergence," Computational Optimization and Applications, Springer, vol. 74(2), pages 583-621, November.
    2. Vasile L. Basescu & John E. Mitchell, 2008. "An Analytic Center Cutting Plane Approach for Conic Programming," Mathematics of Operations Research, INFORMS, vol. 33(3), pages 529-551, August.
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    5. Mohammad R. Oskoorouchi & Hamid R. Ghaffari & Tamás Terlaky & Dionne M. Aleman, 2011. "An Interior Point Constraint Generation Algorithm for Semi-Infinite Optimization with Health-Care Application," Operations Research, INFORMS, vol. 59(5), pages 1184-1197, October.
    6. John Mitchell & Vasile Basescu, 2008. "Selective Gram–Schmidt orthonormalization for conic cutting surface algorithms," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 67(1), pages 91-115, February.

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