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October  2018, 14(4): 1381-1396. doi: 10.3934/jimo.2018012

A power penalty method for a class of linearly constrained variational inequality

Ming Chen 1,, and  Chongchao Huang 2,

1. 

School of Applied Mathematics, Xiamen University of Technology, Xiamen 361024, China
2. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

* Corresponding author: M. Chen

Received  October 2016 Revised  August 2017 Published  October 2018 Early access  January 2018

This paper establishes new convergence results for the power pena-lty method for a mixed complementarity problem(MiCP). The power penalty method approximates the MiCP by a nonlinear equation containing a power penalty term. The main merit of the method is that it has an exponential convergence rate with the penalty parameter when the involved function is continuous and ξ-monotone. Under the same assumptions, we establish a new upper bound for the approximation error of the solution to the nonlinear equation. We also prove that the penalty method can handle general monotone MiCPs. Then the method is used to solve a class of linearly constrained variational inequality(VI). Since the MiCP associated with a linearly constrained VI does not ξ-monotone even if the VI is ξ-monotone, we establish the new convergence result for this MiCP. We use the method to solve the asymmetric traffic assignment problem which can be reformulated as a class of linearly constrained VI. Numerical results are provided to demonstrate the efficiency of the method.

Keywords: Variational inequality, mixed complementarity problem, power penalty method, traffic assignment problem, column generation.
Mathematics Subject Classification: Primary: 90C33, 90B20; Secondary: 90B10.
Citation: Ming Chen, Chongchao Huang. A power penalty method for a class of linearly constrained variational inequality. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1381-1396. doi: 10.3934/jimo.2018012
References:
[1]

M. Chen and C. C. Huang, A power penalty method for the general traffic assignment problem with elastic demand, Journal of Industrial and Management Optimization, 10 (2014), 1019-1030.  doi: 10.3934/jimo.2014.10.1019.  Google Scholar

[2]

F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems Springer-Verlag, New York, 2003. doi: 10.1007/b97543.  Google Scholar

[3]

B. S. He and L. Z. Liao, Improvements of some projection methods for monotone nonlinear variational inequalities, Journal of Optimization Theory & Applications, 112 (2002), 111-128.  doi: 10.1023/A:1013096613105.  Google Scholar

[4]

C. C. Huang and S. Wang, A power penalty approach to a nonlinear complementarity problem, Operations Research Letters, 38 (2010), 72-76.  doi: 10.1016/j.orl.2009.09.009.  Google Scholar

[5]

C. C. Huang and S. Wang, A penalty method for a mixed nonlinear complementarity problem, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), 588-597.  doi: 10.1016/j.na.2011.08.061.  Google Scholar

[6]

S. Lawphongpanich and D. Hearn, Simplical decomposition of the asymmetric traffic assignment problem, Transportation Research Part B: Methodological, 18 (1984), 123-133.  doi: 10.1016/0191-2615(84)90026-2.  Google Scholar

[7]

T. D. Luca, F. Facchinei and C. Kanzow, A semismooth equation approach to the solution of nonlinear complementarity problems, Mathematical Programming, 75 (1996), 407-439.  doi: 10.1007/BF02592192.  Google Scholar

[8]

B. Panicucci, M. Pappalardo and M. Passacantando, A path-based double projection method for solving the asymmetric traffic network equilibrium problem, Optimization Letters, 1 (2007), 171-185.  doi: 10.1007/s11590-006-0002-9.  Google Scholar

[9]

P. Patriksson, The Traffic Assignment Problem: Models and Methods VSP, Utrecht, 1994. Google Scholar

[10]

M. V. Solodov and B. F. Svaiter, A new projection method for variational inequality problems, SIAM Journal on Control & Optimization, 37 (1999), 765-776.  doi: 10.1137/S0363012997317475.  Google Scholar

[11]

K. Taji, M. Fukushima and T. Ibaraki, A globally convergent newton method for solving strongly monotone variational inequalities, Mathematical Programming, 58 (1993), 369-383.  doi: 10.1007/BF01581276.  Google Scholar

[12]

S. Wang, A penalty method for a finite-dimensional obstacle problem with derivative constraints, Optimization Letters, 8 (2014), 1799-1811.  doi: 10.1007/s11590-013-0651-4.  Google Scholar

[13]

S. Wang, A penalty approach to a discretized double obstacle problem with derivative constraints, Journal of Global Optimization, 62 (2015), 775-790.  doi: 10.1007/s10898-014-0262-3.  Google Scholar

[14]

S. Wang and C. S. Huang, A power penalty method for solving a nonlinear parabolic complementarity problem, Nonlinear Analysis: Theory, Methods & Applications, 69 (2008), 1125-1137.  doi: 10.1016/j.na.2007.06.014.  Google Scholar

[15]

S. Wang and X. Q. Yang, A power penalty method for a bounded nonlinear complementarity problem, Optimization: A Journal of Mathematical Programming and Operations Research, 64 (2015), 2377-2394.  doi: 10.1080/02331934.2014.967236.  Google Scholar

[16]

S. Wang and X. Q. Yang, A power penalty method for linear complementarity problems, Operations Research Letters, 36 (2008), 211-214.  doi: 10.1016/j.orl.2007.06.006.  Google Scholar

[17]

S. Wang, X. Q. Yang and K. L. Teo, Power penalty method for a linear complementarity problem arising from American option valuation, Journal of Optimization Theory & Applications, 129 (2006), 227-254.  doi: 10.1007/s10957-006-9062-3.  Google Scholar

show all references

References:
[1]

M. Chen and C. C. Huang, A power penalty method for the general traffic assignment problem with elastic demand, Journal of Industrial and Management Optimization, 10 (2014), 1019-1030.  doi: 10.3934/jimo.2014.10.1019.  Google Scholar

[2]

F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems Springer-Verlag, New York, 2003. doi: 10.1007/b97543.  Google Scholar

[3]

B. S. He and L. Z. Liao, Improvements of some projection methods for monotone nonlinear variational inequalities, Journal of Optimization Theory & Applications, 112 (2002), 111-128.  doi: 10.1023/A:1013096613105.  Google Scholar

[4]

C. C. Huang and S. Wang, A power penalty approach to a nonlinear complementarity problem, Operations Research Letters, 38 (2010), 72-76.  doi: 10.1016/j.orl.2009.09.009.  Google Scholar

[5]

C. C. Huang and S. Wang, A penalty method for a mixed nonlinear complementarity problem, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), 588-597.  doi: 10.1016/j.na.2011.08.061.  Google Scholar

[6]

S. Lawphongpanich and D. Hearn, Simplical decomposition of the asymmetric traffic assignment problem, Transportation Research Part B: Methodological, 18 (1984), 123-133.  doi: 10.1016/0191-2615(84)90026-2.  Google Scholar

[7]

T. D. Luca, F. Facchinei and C. Kanzow, A semismooth equation approach to the solution of nonlinear complementarity problems, Mathematical Programming, 75 (1996), 407-439.  doi: 10.1007/BF02592192.  Google Scholar

[8]

B. Panicucci, M. Pappalardo and M. Passacantando, A path-based double projection method for solving the asymmetric traffic network equilibrium problem, Optimization Letters, 1 (2007), 171-185.  doi: 10.1007/s11590-006-0002-9.  Google Scholar

[9]

P. Patriksson, The Traffic Assignment Problem: Models and Methods VSP, Utrecht, 1994. Google Scholar

[10]

M. V. Solodov and B. F. Svaiter, A new projection method for variational inequality problems, SIAM Journal on Control & Optimization, 37 (1999), 765-776.  doi: 10.1137/S0363012997317475.  Google Scholar

[11]

K. Taji, M. Fukushima and T. Ibaraki, A globally convergent newton method for solving strongly monotone variational inequalities, Mathematical Programming, 58 (1993), 369-383.  doi: 10.1007/BF01581276.  Google Scholar

[12]

S. Wang, A penalty method for a finite-dimensional obstacle problem with derivative constraints, Optimization Letters, 8 (2014), 1799-1811.  doi: 10.1007/s11590-013-0651-4.  Google Scholar

[13]

S. Wang, A penalty approach to a discretized double obstacle problem with derivative constraints, Journal of Global Optimization, 62 (2015), 775-790.  doi: 10.1007/s10898-014-0262-3.  Google Scholar

[14]

S. Wang and C. S. Huang, A power penalty method for solving a nonlinear parabolic complementarity problem, Nonlinear Analysis: Theory, Methods & Applications, 69 (2008), 1125-1137.  doi: 10.1016/j.na.2007.06.014.  Google Scholar

[15]

S. Wang and X. Q. Yang, A power penalty method for a bounded nonlinear complementarity problem, Optimization: A Journal of Mathematical Programming and Operations Research, 64 (2015), 2377-2394.  doi: 10.1080/02331934.2014.967236.  Google Scholar

[16]

S. Wang and X. Q. Yang, A power penalty method for linear complementarity problems, Operations Research Letters, 36 (2008), 211-214.  doi: 10.1016/j.orl.2007.06.006.  Google Scholar

[17]

S. Wang, X. Q. Yang and K. L. Teo, Power penalty method for a linear complementarity problem arising from American option valuation, Journal of Optimization Theory & Applications, 129 (2006), 227-254.  doi: 10.1007/s10957-006-9062-3.  Google Scholar

Figure 1.  Road Network
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Table 1.  Computational Results
O-D pairMinimum costRoute flowRoute cost
1-1222.54047851.126922.540478
170.197822.540478
21.875722.540478
21.551322.540478
35.248322.540478
9-422.006421222.762622.006421
121.421922.006421
55.815522.006421
12-122.59157488.808222.591574
189.444622.591574
18.602922.591574
24.336522.591574
28.807822.591574
4-921.965890195.805121.965890
99.635821.965890
54.559121.965890
Table Options

Download as excel

O-D pairMinimum costRoute flowRoute cost
1-1222.54047851.126922.540478
170.197822.540478
21.875722.540478
21.551322.540478
35.248322.540478
9-422.006421222.762622.006421
121.421922.006421
55.815522.006421
12-122.59157488.808222.591574
189.444622.591574
18.602922.591574
24.336522.591574
28.807822.591574
4-921.965890195.805121.965890
99.635821.965890
54.559121.965890
Table 2.  Computational results when $k = 2$
$\lambda$mhposi-hEtotal costs
5516163.83E-1131159.8244
10516163.83E-1131159.8244
15516163.83E-1131159.8244
20516163.83E-1131159.8244
25516163.83E-1131159.8244
30516163.83E-1131159.8244
35516163.83E-1131159.8244
40516163.83E-1131159.8244
45516163.83E-1131159.8244
50516163.83E-1131159.8244
Table Options

Download as excel

$\lambda$mhposi-hEtotal costs
5516163.83E-1131159.8244
10516163.83E-1131159.8244
15516163.83E-1131159.8244
20516163.83E-1131159.8244
25516163.83E-1131159.8244
30516163.83E-1131159.8244
35516163.83E-1131159.8244
40516163.83E-1131159.8244
45516163.83E-1131159.8244
50516163.83E-1131159.8244
Table 3.  Computational results when $\lambda = 5$
$k$mhposi-hEtotal costs
1516163.83E-1131159.8244
2516163.83E-1131159.8244
3518161.31E-1031159.8244
4617162.43E-1231159.8244
Table Options

Download as excel

$k$mhposi-hEtotal costs
1516163.83E-1131159.8244
2516163.83E-1131159.8244
3518161.31E-1031159.8244
4617162.43E-1231159.8244
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