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Two new predictor-corrector algorithms for second-order cone programming

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Abstract

Based on the ideas of infeasible interior-point methods and predictor-corrector algorithms, two interior-point predictor-corrector algorithms for the second-order cone programming (SOCP) are presented. The two algorithms use the Newton direction and the Euler direction as the predictor directions, respectively. The corrector directions belong to the category of the Alizadeh-Haeberly-Overton (AHO) directions. These algorithms are suitable to the cases of feasible and infeasible interior iterative points. A simpler neighborhood of the central path for the SOCP is proposed, which is the pivotal difference from other interior-point predictor-corrector algorithms. Under some assumptions, the algorithms possess the global, linear, and quadratic convergence. The complexity bound O(rln(ɛ 0/ɛ)) is obtained, where r denotes the number of the second-order cones in the SOCP problem. The numerical results show that the proposed algorithms are effective.

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References

  1. Alizadeh, F. and Goldfarb, D. Second-order cone programming. Mathematical Programming: Series B, 95, 3–51 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Witzgall, C. Optimal Location of a Central Facility, Mathematical Models and Concepts, Technical Report 8388, National Bureau of Standards, Washington, DC (1964)

    Google Scholar 

  3. Lobo, M. S., Vandenberghe, L., Boyd, S., and Lebret, H. Applications of second-order cone programming. Linear Algebra and Its Applications, 284, 193–228 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Kanno, Y., Ohsaki, M., and Ito, J. Large-deformation and friction analysis of non-linear elastic cable networks by second-order cone programming. International Journal for Numerical Methods in Engineering, 55, 1079–1114 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Makrodimopoulos, A. and Martin, C. M. Lower bound limit analysis of cohesive-frictional materials using second-order cone programming. International Journal for Numerical Methods in Engineering, 66, 604–634 (2006)

    Article  MATH  Google Scholar 

  6. Nemirovskii, A. and Scheinberg, K. Extension of Karmarkar’s algorithm onto convex quadratically constrained quadratic problems. Mathematical Programming, 72, 273–289 (1996)

    MATH  MathSciNet  Google Scholar 

  7. Nesterov, Y. E. and Todd, M. J. Self-scaled barriers and interior-point methods for convex programming. Mathematics of Operations Research, 22, 1–42 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Nesterov, Y. E. and Todd, M. J. Primal-dual interior-point methods for self-scaled cones. SIAM Journal on Optimization, 8(2), 324–364 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. Adler, I. and Alizadeh, F. Primal-Dual Interior-Point Algorithms for Convex Quadratically Constrained and Semidefinite Optimization Problems, Technical Report RRR 46-95, RUTCOR, Rutgers University (1995)

  10. Alizadeh, F., Haeberly, J. P. A., and Overton, M. L. Primal-dual interior-point methods for semidefinite programming: convergence rates, stability and numerical results. SIAM Journal on Optimization, 8(3), 746–768 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  11. Monteiro, R. D. C. and Tsuchiya, T. Polynomial convergence of primal-dual algorithms for the second-order cone program based on the MZ-family of directions. Mathematical Programming: Series A, 88, 61–83 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Chi, X. N. and Liu, S. Y. An infeasible-interior-point predictor-corrector algorithm for the second-order cone program. Acta Mathematica Scientia, 28B(3), 551–559 (2008)

    MathSciNet  Google Scholar 

  13. Mizuno, S., Todd, M. J., and Ye, Y. On adaptive-step primal-dual interior-point algorithms for linear programming. Mathematics of Operations Research, 18(4), 964–981 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  14. Miao, J. M. Two infeasible interior-point predictor-corrector algorithms for linear programming. SIAM Journal on Optimization, 6(3), 587–599 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  15. Zhang, Y. On the convergence of a class of infeasible interior-point methods for the horizontal linear complementarity problem. SIAM Journal on Optimization, 4(1), 208–227 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kojima, M. Basic lemmas in polynomial-time infeasible-interior-point methods for linear programs. Annals of Operations Research, 62, 1–28 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  17. Bai, Y. Q., Wang, G. Q., and Roos, C. Primal-dual interior-point algorithms for second-order cone optimization based on kernel functions. Nonlinear Analysis, 70, 3584–3602 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Pataki, G. and Schmieta, S. The DIMACS library of semidefinte-quadratic-linear programs. http://dimacs.rutgers.edu/Challenges/Seventh/Instances/ (2002)

  19. Pan, S. H. and Chen, J. S. A semismooth Newton method for SOCCPs based on a one-parametric class of SOC complementarity functions. Computational Optimization and Applications, 45, 59–88 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, 200444, P. R. China

    You-fang Zeng  (曾有芳) & Yan-qin Bai  (白延琴)

  2. College of Mathematics and Information Science, Guangxi University, Nanning, 530004, P. R. China

    You-fang Zeng  (曾有芳), Jin-bao Jian  (简金宝) & Chun-ming Tang  (唐春明)

Authors
  1. You-fang Zeng  (曾有芳)
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  2. Yan-qin Bai  (白延琴)
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  3. Jin-bao Jian  (简金宝)
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  4. Chun-ming Tang  (唐春明)
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Correspondence to You-fang Zeng  (曾有芳).

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 71061002 and 11071158) and the Natural Science Foundation of Guangxi Province of China (Nos. 0832052 and 2010GXNSFB013047)

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Zeng, Yf., Bai, Yq., Jian, Jb. et al. Two new predictor-corrector algorithms for second-order cone programming. Appl. Math. Mech.-Engl. Ed. 32, 521–532 (2011). https://doi.org/10.1007/s10483-011-1435-x

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  • DOI: https://doi.org/10.1007/s10483-011-1435-x

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