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Selective Gram–Schmidt orthonormalization for conic cutting surface algorithms

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  • John Mitchell
  • Vasile Basescu

Abstract

It is not straightforward to find a new feasible solution when several conic constraints are added to a conic optimization problem. Examples of conic constraints include semidefinite constraints and second order cone constraints. In this paper, a method to slightly modify the constraints is proposed. Because of this modification, a simple procedure to generate strictly feasible points in both the primal and dual spaces can be defined. A second benefit of the modification is an improvement in the complexity analysis of conic cutting surface algorithms. Complexity results for conic cutting surface algorithms proved to date have depended on a condition number of the added constraints. The proposed modification of the constraints leads to a stronger result, with the convergence of the resulting algorithm not dependent on the condition number. Copyright Springer-Verlag 2008

Suggested Citation

  • 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.
    • Handle: RePEc:spr:mathme:v:67:y:2008:i:1:p:91-115
    DOI: 10.1007/s00186-007-0177-6
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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.
    2. J. E. Mitchell & S. Ramaswamy, 2005. "Using Selective Orthonormalization to Update the Analytic Center after Addition of Multiple Cuts," Journal of Optimization Theory and Applications, Springer, vol. 125(2), pages 431-451, May.
    3. 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.
    4. 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.
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