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doi: 10.3934/jimo.2018012

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

 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  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.

Citation: Ming Chen, Chongchao Huang. A power penalty method for a class of linearly constrained variational inequality. Journal of Industrial & Management Optimization, 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. [2] F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems Springer-Verlag, New York, 2003. doi: 10.1007/b97543. [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. [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. [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. [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. [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. [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. [9] P. Patriksson, The Traffic Assignment Problem: Models and Methods VSP, Utrecht, 1994. [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. [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. [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. [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. [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. [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. [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. [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.

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##### 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. [2] F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems Springer-Verlag, New York, 2003. doi: 10.1007/b97543. [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. [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. [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. [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. [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. [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. [9] P. Patriksson, The Traffic Assignment Problem: Models and Methods VSP, Utrecht, 1994. [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. [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. [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. [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. [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. [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. [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. [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.
Computational Results
 O-D pair Minimum cost Route flow Route cost 1-12 22.540478 51.1269 22.540478 170.1978 22.540478 21.8757 22.540478 21.5513 22.540478 35.2483 22.540478 9-4 22.006421 222.7626 22.006421 121.4219 22.006421 55.8155 22.006421 12-1 22.591574 88.8082 22.591574 189.4446 22.591574 18.6029 22.591574 24.3365 22.591574 28.8078 22.591574 4-9 21.965890 195.8051 21.965890 99.6358 21.965890 54.5591 21.965890
 O-D pair Minimum cost Route flow Route cost 1-12 22.540478 51.1269 22.540478 170.1978 22.540478 21.8757 22.540478 21.5513 22.540478 35.2483 22.540478 9-4 22.006421 222.7626 22.006421 121.4219 22.006421 55.8155 22.006421 12-1 22.591574 88.8082 22.591574 189.4446 22.591574 18.6029 22.591574 24.3365 22.591574 28.8078 22.591574 4-9 21.965890 195.8051 21.965890 99.6358 21.965890 54.5591 21.965890
Computational results when $k = 2$
 $\lambda$ m h posi-h E total costs 5 5 16 16 3.83E-11 31159.8244 10 5 16 16 3.83E-11 31159.8244 15 5 16 16 3.83E-11 31159.8244 20 5 16 16 3.83E-11 31159.8244 25 5 16 16 3.83E-11 31159.8244 30 5 16 16 3.83E-11 31159.8244 35 5 16 16 3.83E-11 31159.8244 40 5 16 16 3.83E-11 31159.8244 45 5 16 16 3.83E-11 31159.8244 50 5 16 16 3.83E-11 31159.8244
 $\lambda$ m h posi-h E total costs 5 5 16 16 3.83E-11 31159.8244 10 5 16 16 3.83E-11 31159.8244 15 5 16 16 3.83E-11 31159.8244 20 5 16 16 3.83E-11 31159.8244 25 5 16 16 3.83E-11 31159.8244 30 5 16 16 3.83E-11 31159.8244 35 5 16 16 3.83E-11 31159.8244 40 5 16 16 3.83E-11 31159.8244 45 5 16 16 3.83E-11 31159.8244 50 5 16 16 3.83E-11 31159.8244
Computational results when $\lambda = 5$
 $k$ m h posi-h E total costs 1 5 16 16 3.83E-11 31159.8244 2 5 16 16 3.83E-11 31159.8244 3 5 18 16 1.31E-10 31159.8244 4 6 17 16 2.43E-12 31159.8244
 $k$ m h posi-h E total costs 1 5 16 16 3.83E-11 31159.8244 2 5 16 16 3.83E-11 31159.8244 3 5 18 16 1.31E-10 31159.8244 4 6 17 16 2.43E-12 31159.8244
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