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2022, Volume 30, Issue 4: 1454-1462. doi: 10.3934/era.2022076
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The parameter-Newton iteration for the second-order cone linear complementarity problem

  • 1.
    School of Mathematics and Computer Sciences, Nanchang University, China
  • 2.
    Jiangxi Institute of Economic Administrators, China
  • Received: 08 December 2021 Revised: 27 February 2022 Accepted: 06 March 2022 Published: 21 March 2022

  • In this paper, we propose the parameter-Newton (PN) method to solve the second-order linear complementarity problem (SOCLCP). The key idea of PN method is that we transfer the SOCLCP into a system of nonlinear equations by bringing in a parameter. Then we solve the system of nonlinear equations by Newton method. At last, we prove that the PN method has quadratic convergence. Compared with the bisection-Newton (BN) method, the PN method has less CPU time and higher accuracy in numerical tests.

    Citation: Peng Zhou, Teng Wang. The parameter-Newton iteration for the second-order cone linear complementarity problem[J]. Electronic Research Archive, 2022, 30(4): 1454-1462. doi: 10.3934/era.2022076

    Related Papers:

  • In this paper, we propose the parameter-Newton (PN) method to solve the second-order linear complementarity problem (SOCLCP). The key idea of PN method is that we transfer the SOCLCP into a system of nonlinear equations by bringing in a parameter. Then we solve the system of nonlinear equations by Newton method. At last, we prove that the PN method has quadratic convergence. Compared with the bisection-Newton (BN) method, the PN method has less CPU time and higher accuracy in numerical tests.


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    [1] A. Alfred, Variational inequalities over the cone of semidefinite positive symmetric matrices and over the Lorentz cone, Optim. Method Softw., 18 (2003), 359–376. https://doi.org/10.1080/1055678031000122586 doi: 10.1080/1055678031000122586
    [2] M. Kojima, S. Shindoh, S. Hara, Interior-point methods for the monotone semidfinite linear complementarity problem in symmetric matrices, SIAM J. Optim., 7 (1997), 86–125. https://doi.org/10.1137/S1052623494269035 doi: 10.1137/S1052623494269035
    [3] A. Yoshise, Interior point trajectories and a homogeneous model for nonlinear complementarity problems over symmetric cones, SIAM J. Optim., 17 (2006), 1129–1153. https://doi.org/10.1137/04061427X doi: 10.1137/04061427X
    [4] S. Hayashi, N. Yamashita, M. Fukushima, Robust Nash equilibria and second-order cone complementarity problems, J. Nonlinear Convex Anal., 6 (2005), 283–296.
    [5] X. D. Chen, D. 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] S. L. Miguel, V. Lieven, B. Stephen, L. Herve, Applications of second-order cone programming, Linear Algebra Appl., 284 (1998), 193–228. https://doi.org/10.1016/S0024-3795(98)10032-0 doi: 10.1016/S0024-3795(98)10032-0
    [7] P. K. Shivaswamy, C. Bhattacharyya, A. J. Smola, Second order cone programming approaches for handling missing and uncertain data, J. Mach. Learn. Res., 7 (2006), 1283–1314.
    [8] S. Kim, M. Kojima, M. Yamashita, Second order cone programming relaxation of a positive semidefinite constraint, Optim. Method Software, 18 (2003), 535–541. https://doi.org/10.1080/1055678031000148696 doi: 10.1080/1055678031000148696
    [9] T. Sasakawa, T. Tsuchiya, Optimal magnetic shield design with second-order cone programming, SIAM J. Sci. Comput., 24 (2003), 1930–1950. https://doi.org/10.1137/S1064827500380350 doi: 10.1137/S1064827500380350
    [10] S. Yan, Y. L. Ma, Optimal design and verification of temporal and spatial filters using second-order cone programming approach, Sci. China Ser. F Inf. Sci., 49 (2006), 235–253. https://doi.org/10.1007/s11432-006-0235-3 doi: 10.1007/s11432-006-0235-3
    [11] Y. Kanno, M. Ohsaki, Contact analysis of cable networks by using second-order cone programming, SIAM J. Sci. Comput., 27 (2006), 2032–2052. https://doi.org/10.1137/S1064827503431946 doi: 10.1137/S1064827503431946
    [12] X. Peng, I. King, Robust BMPM training based on second-order cone programming and its application in medical diagnosis, Neural Netw., 21 (2008), 450–457. https://doi.org/10.1016/j.neunet.2007.12.051 doi: 10.1016/j.neunet.2007.12.051
    [13] D. Goldfarb, W. T. Yin, Second-order cone programming methods for total variation-based image restoration, SIAM J. Sci. Comput., 27 (2005), 622–645. https://doi.org/10.1137/040608982 doi: 10.1137/040608982
    [14] S. Yan, Y. L. Ma, Robust supergain beamforming for circular array via second order cone programm, Appl. Acoust., 66 (2005), 1018–1032. https://doi.org/10.1016/j.apacoust.2005.01.003 doi: 10.1016/j.apacoust.2005.01.003
    [15] M. Fukushima, Z. Q. Luo, P. Tseng, Smoothing functions for second-order cone complementarity problems, SIAM J. Optim., 12 (2001), 436–460. https://doi.org/10.1137/S1052623400380365 doi: 10.1137/S1052623400380365
    [16] M. S. Gowda, Y. Song, On semidfinite linear complementarity problems, Math. Program., 88 (2000), 575–587. https://doi.org/10.1007/PL00011387 doi: 10.1007/PL00011387
    [17] M. S. Gowda, R. Sznajder, Automorphism invariance of P-matrix and GUS properties of linear transformations on Euclidean Jordan algebras, Math. Oper. Res., 31 (2006), 109–123. https://doi.org/10.1287/moor.1050.0182 doi: 10.1287/moor.1050.0182
    [18] W. H. Yang, Y. Yuan, The GUS-property of second-order cone linear complementarity problems, Math. Comput., 141 (2013), 295–317. https://doi.org/10.1007/s10107-012-0523-1 doi: 10.1007/s10107-012-0523-1
    [19] 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
    [20] S. Z. Xiang, Y. L. San, H. L. Zhen, A smoothing method for second order cone complementarity problem, J. Comput. Appl. Math., 228 (2009), 83–91. https://doi.org/10.1016/j.cam.2008.08.040 doi: 10.1016/j.cam.2008.08.040
    [21] B. Kheirfam, N. M. Amiri, A new interior-point algorithm based on modified Nesterov-Todd direction for symmetric cone linear complementarity problem, Optim. Lett., 8 (2014), 1017–1029. https://doi.org/10.1007/s11590-013-0618-5 doi: 10.1007/s11590-013-0618-5
    [22] G. Q. Wang, D. T. Zhu, A class of polynomial interior-point algorithms for the Cartesian $P_{*}(\kappa)$ second-order cone linear complementarity problem, Nonlinear Anal., Theory, Methods Appl., 73 (2010), 3705–3722. https://doi.org/10.1007/s10957-011-9938-8 doi: 10.1007/s10957-011-9938-8
    [23] G. Q. Wang, Y. Q. Bai, A class of polynomial interior-point algorithms for the Cartesian P-matrix linear complementarity problem over symmetric cones, J. Optim. Theory Appl., 152 (2012), 739–772. https://doi.org/10.1007/s10957-011-9938-8 doi: 10.1007/s10957-011-9938-8
    [24] G. Lesaja, G. Q. Wang, D. T. Zhu, Interior-point methods for Cartesian $P_{*}(\kappa)$-linear complementarity problems over symmetric cones based on the eligible kernel functions, Optim. Methods Software, 27 (2012), 827–843. https://doi.org/10.1080/10556788.2012.670858 doi: 10.1080/10556788.2012.670858
    [25] G. Q. Wang, G. Lesaja, Full Nesterov-Todd step feasible interior-point method for the Cartesian $P_{*}(\kappa)$-SCLCP, Optim. Methods Software, 28 (2013), 600–618. https://doi.org/10.1080/10556788.2013.781600 doi: 10.1080/10556788.2013.781600
    [26] L. Fang, C. Y. Han, A new one-step smoothing newton method for the second-order cone complementarity problem, Math. Meth. Appl. Sci., 34 (2011), 347–359. https://doi.org/10.1002/mma.1366 doi: 10.1002/mma.1366
    [27] J. Y. Tang, G. P. He, D. Li, F. Liang, J. C. Zhou, A smoothing Newton method for the second-order cone complementarity problem, J. Optim. Theory Appl., 58 (2013), 223–247. https://doi.org/10.1007/s10492-013-0011-9 doi: 10.1007/s10492-013-0011-9
    [28] L. H Zhang, W. H. Yang, An efficient algorithm for second-order cone linear complementarity problems, Math. Comput., 83 (2013), 1701–1726. https://doi.org/10.1090/S0025-5718-2013-02795-0 doi: 10.1090/S0025-5718-2013-02795-0
    [29] Z. J. Hao, Z. P. Wan, X. N. Chi, A power penalty method for second-order cone linear complementarity problems, Oper. Res. Lett., 43 (2015), 137–142. https://doi.org/10.1016/j.orl.2014.12.012 doi: 10.1016/j.orl.2014.12.012
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