April  2017, 13(2): 531-548. doi: 10.3934/jimo.2016030

The linear convergence of a derivative-free descent method for nonlinear complementarity problems

1. 

Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, China

2. 

Department of Mathematics, School of Science, Tianjin University of Technology, Tianjin 300384, China

* Corresponding author: Li-Yong Lu

Received  November 2012 Revised  March 2016 Published  May 2016

Recently, Hu, Huang and Chen [Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems, J. Comput. Appl. Math. 230 (2009): 69-82] introduced a family of generalized NCP-functions, which include many existing NCP-functions as special cases. They obtained several favorite properties of the functions; and by which, they showed that a derivative-free descent method is globally convergent under suitable assumptions. However, no result on convergent rate of the method was reported. In this paper, we further investigate some properties of this family of generalized NCP-functions. In particular, we show that, under suitable assumptions, the iterative sequence generated by the descent method discussed in their paper converges globally at a linear rate to a solution of the nonlinear complementarity problem. Some preliminary numerical results are reported, which verify the theoretical results obtained.

Citation: Wei-Zhe Gu, Li-Yong Lu. The linear convergence of a derivative-free descent method for nonlinear complementarity problems. Journal of Industrial and Management Optimization, 2017, 13 (2) : 531-548. doi: 10.3934/jimo.2016030
References:
[1]

S. C. BillupsS. P. Drikse and M. C. Soares, A comparison of algorithm for large scale mixed complementartiy problems, Comput. Optim. Appl., 7 (1977), 3-25.  doi: 10.1023/A:1008632215341.

[2]

J.-S. Chen, The semismooth-related properties of a merit function and a descent method for the nonlinear complementarity problem, J. Global Optim., 36 (2006), 565-580.  doi: 10.1007/s10898-006-9027-y.

[3]

J.-S. ChenZ. H. Huang and C.-Y. She, A new class of penalized NCP-functions and its properties, Comput. Optim. Appl., 50 (2011), 49-73.  doi: 10.1007/s10589-009-9315-9.

[4]

J.-S. Chen and S. H. Pan, A regularization semismooth Newton method based on the generalized Fischer-Burmeister function for $P_0$-NCPs, J. Comput. Appl. Math., 220 (2008), 464-479.  doi: 10.1016/j.cam.2007.08.020.

[5]

J.-S. Chen and S. H. Pan, A family of NCP-functions and a descent method for the nonlinear complementarity problem, Comput. Optim. Appl., 40 (2008), 389-404.  doi: 10.1007/s10589-007-9086-0.

[6]

J.-S. ChenH.-T. Gao and S. H. Pan, An $R$-linearly convergent derivative-free algorithm for nonlinear complementarity problems based on the generalized Fisher-Burmeister merit function, Comput. Optim. Appl., 232 (2009), 455-471.  doi: 10.1016/j.cam.2009.06.022.

[7]

F. Facchinei and J. S. Pang, Finite-dimensional variational inequalities and complementarity problems, Springer Verlag, New York, 2003.

[8]

F. Facchinei and J. Soares, A new merit function for nonlinear complementarity problems and a related algorithm, SIAM J. Optim., 7 (1997), 225-247.  doi: 10.1137/S1052623494279110.

[9]

M. C. Ferris and J. S. Pang, Engineering and economic applications of complementarity problems, SIAM Rev., 39 (1997), 669-713.  doi: 10.1137/S0036144595285963.

[10]

C. Geiger and C. Kanzow, On the resolution of monotone complementarity problems, Comput. Optim. Appl., 5 (1996), 155-173.  doi: 10.1007/BF00249054.

[11]

P. T. Harker and J. S. Pang, Finite dimensional variational inequality and nonlinear complementarity problem: A survey of theory, algorithms and applications, Math. Program., 48 (1990), 161-220.  doi: 10.1007/BF01582255.

[12]

S. L. HuZ. H. Huang and J.-S. Chen, Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems, J. Comput. Appl. Math., 230 (2009), 69-82.  doi: 10.1016/j.cam.2008.10.056.

[13]

S. L. HuZ. H. Huang and N. Lu, Smoothness of a class of generalized merit functions for the second-order cone complementarity problem, Pacific J. Optim., 6 (2010), 551-571. 

[14]

Z. H. Huang, The global linear and local quadratic convergence of a non-interior continuation algorithm for the LCP, IMA J. Numer. Anal., 25 (2005), 670-684.  doi: 10.1093/imanum/dri008.

[15]

Z. H. HuangL. Qi and D. Sun, Sub-quadratic convergence of a smoothing Newton algorithm for the $P_0$-and monotone LCP, Math. Program., 99 (2004), 423-441.  doi: 10.1007/s10107-003-0457-8.

[16]

H. Y. JiangM. Fukushima and L. Qi, A trust region method for solving generalized complementarity problems, SIAM. J. Optim., 8 (1998), 140-157.  doi: 10.1137/S1052623495296541.

[17]

C. Kanzow and H. Kleinmichel, A new class of semismooth Newton method for nonlinear complementarity problems, Comput. Optim. Appl., 11 (1998), 227-251.  doi: 10.1023/A:1026424918464.

[18]

L. Y. LuZ. H. Huang and S. L. Hu, Properties of a family of merit functions and a merit function method for the NCP, Appl. Math. – J. Chinese Univ., 25 (2010), 379-390.  doi: 10.1007/s11766-010-2179-z.

[19]

Z. Q. Luo and P. Tseng, A new class of merit functions for the nonlinear complementarity problem, in Complementarity and Variational Problems: State of the Art, eds. M. C. Ferris and J. -S. Pang, SIAM, Philadelphia, (1997), 204–225.

[20]

O. L. Mangasarian and M. V. Solodov, A linearly convergent derivative-free descent method for strongly monotone complementarity problems, Comput. Optim. Appl., 14 (1999), 5-16.  doi: 10.1023/A:1008752626695.

[21]

J. S. Pang, A posteriori error bounds for the linearly-constrained variational inequality problem, Math. Oper. Res., 12 (1987), 474-484.  doi: 10.1287/moor.12.3.474.

[22]

J. M. Ortega and W. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719468.

[23]

L. QiD. Sun and L. 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. 

[24]

K. Yamada, N. Yamashita and M. Fukushima, A new derivative-free descent method for the nonlinear complementarity problem, in Nonlinear Optimization and Related Topics(eds. G. D. Pillo and F. Giannessi), Kluwer Academic, Dordrecht, 36 (2000), 463–489. doi: 10.1007/978-1-4757-3226-9_25.

show all references

References:
[1]

S. C. BillupsS. P. Drikse and M. C. Soares, A comparison of algorithm for large scale mixed complementartiy problems, Comput. Optim. Appl., 7 (1977), 3-25.  doi: 10.1023/A:1008632215341.

[2]

J.-S. Chen, The semismooth-related properties of a merit function and a descent method for the nonlinear complementarity problem, J. Global Optim., 36 (2006), 565-580.  doi: 10.1007/s10898-006-9027-y.

[3]

J.-S. ChenZ. H. Huang and C.-Y. She, A new class of penalized NCP-functions and its properties, Comput. Optim. Appl., 50 (2011), 49-73.  doi: 10.1007/s10589-009-9315-9.

[4]

J.-S. Chen and S. H. Pan, A regularization semismooth Newton method based on the generalized Fischer-Burmeister function for $P_0$-NCPs, J. Comput. Appl. Math., 220 (2008), 464-479.  doi: 10.1016/j.cam.2007.08.020.

[5]

J.-S. Chen and S. H. Pan, A family of NCP-functions and a descent method for the nonlinear complementarity problem, Comput. Optim. Appl., 40 (2008), 389-404.  doi: 10.1007/s10589-007-9086-0.

[6]

J.-S. ChenH.-T. Gao and S. H. Pan, An $R$-linearly convergent derivative-free algorithm for nonlinear complementarity problems based on the generalized Fisher-Burmeister merit function, Comput. Optim. Appl., 232 (2009), 455-471.  doi: 10.1016/j.cam.2009.06.022.

[7]

F. Facchinei and J. S. Pang, Finite-dimensional variational inequalities and complementarity problems, Springer Verlag, New York, 2003.

[8]

F. Facchinei and J. Soares, A new merit function for nonlinear complementarity problems and a related algorithm, SIAM J. Optim., 7 (1997), 225-247.  doi: 10.1137/S1052623494279110.

[9]

M. C. Ferris and J. S. Pang, Engineering and economic applications of complementarity problems, SIAM Rev., 39 (1997), 669-713.  doi: 10.1137/S0036144595285963.

[10]

C. Geiger and C. Kanzow, On the resolution of monotone complementarity problems, Comput. Optim. Appl., 5 (1996), 155-173.  doi: 10.1007/BF00249054.

[11]

P. T. Harker and J. S. Pang, Finite dimensional variational inequality and nonlinear complementarity problem: A survey of theory, algorithms and applications, Math. Program., 48 (1990), 161-220.  doi: 10.1007/BF01582255.

[12]

S. L. HuZ. H. Huang and J.-S. Chen, Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems, J. Comput. Appl. Math., 230 (2009), 69-82.  doi: 10.1016/j.cam.2008.10.056.

[13]

S. L. HuZ. H. Huang and N. Lu, Smoothness of a class of generalized merit functions for the second-order cone complementarity problem, Pacific J. Optim., 6 (2010), 551-571. 

[14]

Z. H. Huang, The global linear and local quadratic convergence of a non-interior continuation algorithm for the LCP, IMA J. Numer. Anal., 25 (2005), 670-684.  doi: 10.1093/imanum/dri008.

[15]

Z. H. HuangL. Qi and D. Sun, Sub-quadratic convergence of a smoothing Newton algorithm for the $P_0$-and monotone LCP, Math. Program., 99 (2004), 423-441.  doi: 10.1007/s10107-003-0457-8.

[16]

H. Y. JiangM. Fukushima and L. Qi, A trust region method for solving generalized complementarity problems, SIAM. J. Optim., 8 (1998), 140-157.  doi: 10.1137/S1052623495296541.

[17]

C. Kanzow and H. Kleinmichel, A new class of semismooth Newton method for nonlinear complementarity problems, Comput. Optim. Appl., 11 (1998), 227-251.  doi: 10.1023/A:1026424918464.

[18]

L. Y. LuZ. H. Huang and S. L. Hu, Properties of a family of merit functions and a merit function method for the NCP, Appl. Math. – J. Chinese Univ., 25 (2010), 379-390.  doi: 10.1007/s11766-010-2179-z.

[19]

Z. Q. Luo and P. Tseng, A new class of merit functions for the nonlinear complementarity problem, in Complementarity and Variational Problems: State of the Art, eds. M. C. Ferris and J. -S. Pang, SIAM, Philadelphia, (1997), 204–225.

[20]

O. L. Mangasarian and M. V. Solodov, A linearly convergent derivative-free descent method for strongly monotone complementarity problems, Comput. Optim. Appl., 14 (1999), 5-16.  doi: 10.1023/A:1008752626695.

[21]

J. S. Pang, A posteriori error bounds for the linearly-constrained variational inequality problem, Math. Oper. Res., 12 (1987), 474-484.  doi: 10.1287/moor.12.3.474.

[22]

J. M. Ortega and W. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719468.

[23]

L. QiD. Sun and L. 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. 

[24]

K. Yamada, N. Yamashita and M. Fukushima, A new derivative-free descent method for the nonlinear complementarity problem, in Nonlinear Optimization and Related Topics(eds. G. D. Pillo and F. Giannessi), Kluwer Academic, Dordrecht, 36 (2000), 463–489. doi: 10.1007/978-1-4757-3226-9_25.

Figure 1.  Convergence behavior of "gafni(1)" with p = 1.1
Figure 2.  Convergence behavior of "gafni(1)" with p = 10
Figure 3.  Convergence behavior of "gafni(1)" with p = 100
Figure 4.  Convergence behavior of "josephy(5)" with θ = 0
Figure 5.  Convergence behavior of "josephy(5)" with θ = 0.5
Figure 6.  Convergence behavior of "josephy(5)" with θ = 1
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