Search Advanced

AIMS Mathematics

2022, Volume 7, Issue 9: 17612-17627. doi: 10.3934/math.2022970
Previous Article Next Article
Research article

New convergence analysis of a class of smoothing Newton-type methods for second-order cone complementarity problem

  • College of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China
  • Received: 30 May 2022 Revised: 12 July 2022 Accepted: 21 July 2022 Published: 01 August 2022

  • MSC : 65K05, 90C33

  • In this paper we propose a class of smoothing Newton-type methods for solving the second-order cone complementarity problem (SOCCP). The proposed method design is based on a special regularized Chen-Harker-Kanzow-Smale (CHKS) smoothing function. When the solution set of the SOCCP is nonempty, our method has the following convergence properties: (ⅰ) it generates a bounded iteration sequence; (ⅱ) the value of the merit function converges to zero; (ⅲ) any accumulation point of the generated iteration sequence is a solution of the SOCCP; (ⅳ) it has the local quadratic convergence rate under suitable assumptions. Some numerical results are reported.

    Citation: Li Dong, Jingyong Tang. New convergence analysis of a class of smoothing Newton-type methods for second-order cone complementarity problem[J]. AIMS Mathematics, 2022, 7(9): 17612-17627. doi: 10.3934/math.2022970

    Related Papers:

  • In this paper we propose a class of smoothing Newton-type methods for solving the second-order cone complementarity problem (SOCCP). The proposed method design is based on a special regularized Chen-Harker-Kanzow-Smale (CHKS) smoothing function. When the solution set of the SOCCP is nonempty, our method has the following convergence properties: (ⅰ) it generates a bounded iteration sequence; (ⅱ) the value of the merit function converges to zero; (ⅲ) any accumulation point of the generated iteration sequence is a solution of the SOCCP; (ⅳ) it has the local quadratic convergence rate under suitable assumptions. Some numerical results are reported.


    加载中


    [1] L. J. Chen, C. F. Ma, A modified smoothing and regularized Newton method for monotone second-order cone complementarity problems, Comput. Math. Appl., 61 (2011), 1047–1418. https://doi.org/10.1016/j.camwa.2011.01.009 doi: 10.1016/j.camwa.2011.01.009
    [2] J. S. Chen, Two classes of merit functions for the second-order cone complementarity problem, Math. Meth. Oper. Res., 64 (2006), 495–519. https://doi.org/10.1007/s00186-006-0098-9 doi: 10.1007/s00186-006-0098-9
    [3] J. S. Chen, S. H. Pan, A survey on SOC complementarity functions and solution methods for SOCPs and SOCCPs, Pac. J. Optim., 8 (2012), 33–74.
    [4] J. S. Chen, S. H. Pan, A descent method for a reformulation of the second-order cone complementarity problem, J. Comput. Appl. Math., 213 (2008), 547–558. https://doi.org/10.1016/j.cam.2007.01.029 doi: 10.1016/j.cam.2007.01.029
    [5] X. D. Chen, D. F. Sun, J. Sun, Complementarity functions and numerical experiments on some smoothing Newton methods for second-order-cone complementarity problems, Comput. Optim. Appl., 25 (2003), 39–56. https://doi.org/10.1023/A:1022996819381 doi: 10.1023/A:1022996819381
    [6] X. N. Chi, S. Y. Liu, A non-interior continuation method for second-order cone optimization, Optimization, 58 (2009), 965–979. https://doi.org/10.1080/02331930701763421 doi: 10.1080/02331930701763421
    [7] L. Dong, J. Y. Tang, J. C. Zhou, A smoothing Newton algorithm for solving the monotone second-order cone complementarity problems, J. Appl. Math. Comput., 40 (2012), 45–61. https://doi.org/10.1007/s12190-012-0550-3 doi: 10.1007/s12190-012-0550-3
    [8] M. Fukushima, Z. Luo, P. Tseng, Smoothing functions for second-order-cone complementarity problems, SIAM J. Optim., 12 (2002), 436–460. https://doi.org/10.1137/S1052623400380365 doi: 10.1137/S1052623400380365
    [9] S. Hayashi, N. Yamashita, M. Fukushima, A combined smoothing and regularization method for monotone second-order cone complementarity problems, SIAM J. Optim., 15 (2005), 593–615. https://doi.org/10.1137/S1052623403421516 doi: 10.1137/S1052623403421516
    [10] Z. H. Huang, S. L. Hu, J. Y. Han, Convergence of a smoothing algorithm for symmetric cone complementarity problems with a nonmonotone line search, Sci. China Ser. A-Math., 52 (2009), 833–848. https://doi.org/10.1007/s11425-008-0170-4 doi: 10.1007/s11425-008-0170-4
    [11] N. Huang, C. F. Ma, A regularized smoothing Newton method for solving SOCCPs based on a new smoothing C-function, Appl. Math. Comput., 230 (2014), 315–329. https://doi.org/10.1016/j.amc.2013.12.116 doi: 10.1016/j.amc.2013.12.116
    [12] Z. H. Huang, T. Ni, Smoothing algorithms for complementarity problems over symmetric cones, Comput. Optim. Appl., 45 (2010), 557–579. https://doi.org/10.1007/s10589-008-9180-y doi: 10.1007/s10589-008-9180-y
    [13] H. Jiang, Smoothed Fischer-Burmeister equation methods for the complementarity problem, Department of Mathematics, The University of Melbourne, Parille, Victoria, Australia, Technical Report, June, 1997.
    [14] Y. Kanno, J. Martins, A. Costa, Three-dimensional quasi-static frictional contact by using second-order cone linear complementarity problem, Int. J. Numer. Meth. Eng., 65 (2006), 62–83. https://doi.org/10.1002/nme.1493 doi: 10.1002/nme.1493
    [15] L. X. Liu, S. Y. Liu, A smoothing Newton method based on a one-parametric class of smoothing function for SOCCP, J. Appl. Math. Comput., 36 (2011), 473–487. https://doi.org/10.1007/s12190-010-0414-7 doi: 10.1007/s12190-010-0414-7
    [16] N. Lu, Z. H. Huang, Solvability of Newton equations in smoothing-type algorithms for the SOCCP, J. Comput. Appl. Math., 235 (2011), 2270–2276. https://doi.org/10.1016/j.cam.2010.10.025 doi: 10.1016/j.cam.2010.10.025
    [17] Y. Narushima, N. Sagara, H. Ogasawara, A smoothing Newton method with Fischer-Burmeister function for second-order cone complementarity problems, J. Optim. Theory Appl., 149 (2011), 79–101. https://doi.org/10.1007/s10957-010-9776-0 doi: 10.1007/s10957-010-9776-0
    [18] S. H. Pan, J. S. Chen, A regularization method for the second-order cone complementarity problem with the Cartesian $P_0$-property, Nonlinear Anal. Theor., 70 (2009), 1475–1491. https://doi.org/10.1016/j.na.2008.02.028 doi: 10.1016/j.na.2008.02.028
    [19] L. Qi, D. Sun, G. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequality problems, Math. Program., 87 (2000), 1–35. https://doi.org/10.1007/s101079900127 doi: 10.1007/s101079900127
    [20] S. P. Rui, C. X. Xu, An inexact smoothing method for SOCCPs based on a one-parametric class of smoothing function, Appl. Math. Comput., 241 (2014), 167–182. https://doi.org/10.1016/j.amc.2014.05.007 doi: 10.1016/j.amc.2014.05.007
    [21] J. Y. Tang, L. Dong, J. C. Zhou, L. Sun, A smoothing-type algorithm for the second-order cone complementarity problem with a new nonmonotone line search, Optimization, 64 (2015), 1935–1955. https://doi.org/10.1080/02331934.2014.906595 doi: 10.1080/02331934.2014.906595
    [22] J. Y. Tang, J. C. Zhou, L. Fang, A non-monotone regularization Newton method for the second-order cone complementarity problem, Appl. Math. Comput., 271 (2015), 743–756. https://doi.org/10.1016/j.amc.2015.09.017 doi: 10.1016/j.amc.2015.09.017
    [23] X. S. Zhang, S. Y. Liu, Z. H. Liu, A regularization smoothing method for second-order cone complementarity problem, Nonlinear Anal. Real, 12 (2011), 731–740. https://doi.org/10.1016/j.nonrwa.2010.08.001 doi: 10.1016/j.nonrwa.2010.08.001
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
AIMS Mathematics
1.8 3.4

Metrics

Article views(834) PDF downloads(38) Cited by(0)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(4)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog