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Polynomial Convergence of Second-Order Mehrotra-Type Predictor-Corrector Algorithms over Symmetric Cones

Author

Listed:
  • Changhe Liu

    (Xidian University
    Henan University of Science and Technology)

  • Hongwei Liu

    (Xidian University)

  • Xinze Liu

    (Xidian University
    Lincang Teachers’ College)

Abstract

This paper presents an extension of the variant of Mehrotra’s predictor–corrector algorithm which was proposed by Salahi and Mahdavi-Amiri (Appl. Math. Comput. 183:646–658, 2006) for linear programming to symmetric cones. This algorithm incorporates a safeguard in Mehrotra’s original predictor–corrector algorithm, which keeps the iterates in the prescribed neighborhood and allows us to get a reasonably large step size. In our algorithm, the safeguard strategy is not necessarily used when the affine scaling step behaves poorly, which is different from the algorithm of Salahi and Mahdavi-Amiri. We slightly modify the maximum step size in the affine scaling step and extend the algorithm to symmetric cones using the machinery of Euclidean Jordan algebras. Based on the Nesterov–Todd direction, we show that the iteration-complexity bound of the proposed algorithm is $\mathcal{O}(r\log\varepsilon^{-1})$ , where r is the rank of the associated Euclidean Jordan algebras and ε>0 is the required precision.

Suggested Citation

  • Changhe Liu & Hongwei Liu & Xinze Liu, 2012. "Polynomial Convergence of Second-Order Mehrotra-Type Predictor-Corrector Algorithms over Symmetric Cones," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 949-965, September.
    • Handle: RePEc:spr:joptap:v:154:y:2012:i:3:d:10.1007_s10957-012-0018-5
    DOI: 10.1007/s10957-012-0018-5
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    References listed on IDEAS

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    1. Yu. E. Nesterov & M. J. Todd, 1997. "Self-Scaled Barriers and Interior-Point Methods for Convex Programming," Mathematics of Operations Research, INFORMS, vol. 22(1), pages 1-42, February.
    2. Jian Zhang & Kecun Zhang, 2011. "Polynomial complexity of an interior point algorithm with a second order corrector step for symmetric cone programming," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 73(1), pages 75-90, February.
    3. Gu, G. & Zangiabadi, M. & Roos, C., 2011. "Full Nesterov-Todd step infeasible interior-point method for symmetric optimization," European Journal of Operational Research, Elsevier, vol. 214(3), pages 473-484, November.
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    Cited by:

    1. Ximei Yang & Hongwei Liu & Yinkui Zhang, 2015. "A New Strategy in the Complexity Analysis of an Infeasible-Interior-Point Method for Symmetric Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 572-587, August.
    2. G. Q. Wang & L. C. Kong & J. Y. Tao & G. Lesaja, 2015. "Improved Complexity Analysis of Full Nesterov–Todd Step Feasible Interior-Point Method for Symmetric Optimization," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 588-604, August.
    3. Hongwei Liu & Ximei Yang & Changhe Liu, 2013. "A New Wide Neighborhood Primal–Dual Infeasible-Interior-Point Method for Symmetric Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 158(3), pages 796-815, September.

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