# American Institute of Mathematical Sciences

January  2011, 7(1): 183-198. doi: 10.3934/jimo.2011.7.183

## A differential equation method for solving box constrained variational inequality problems

 1 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China 2 School of Sciences, Dalian Nationalities University, Dalian, 116066, China 3 Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, LiaoNing

Received  April 2010 Revised  October 2010 Published  January 2011

In this paper, we discuss a system of differential equations based on the projection operator for solving the box constrained variational inequality problems. The equilibrium solutions to the differential equation system are proved to be the solutions of the box constrained variational inequality problems. Two differential inclusion problems associated with the system of differential equations are introduced. It is proved that the equilibrium solution to the differential equation system is locally asymptotically stable by verifying the locally asymptotical stability of the equilibrium positions of the differential inclusion problems. An Euler discrete scheme with Armijo line search rule is introduced and its global convergence is demonstrated. The numerical experiments are reported to show that the Euler method is effective.
Citation: Li Wang, Yang Li, Liwei Zhang. A differential equation method for solving box constrained variational inequality problems. Journal of Industrial and Management Optimization, 2011, 7 (1) : 183-198. doi: 10.3934/jimo.2011.7.183
##### References:
 [1] K. J. Arrow and L. Hurwicz, Reduction of constrained maxima to saddle point problems, in "Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability" (J. Neyman Ed.), University of California Press, Berkeleyand Los Angeles, 5 (1956), 1-20. [2] J. Chen, C. Ko and S. Pan, A neural network based on the generalized Fischer-Burmeister function for nonlinear complementarity problems, Information Sciences, 180 (1992), 697-711. doi: 10.1016/j.ins.2009.11.014. [3] C. Dang, Y. Leung, X. Gao and K. Chen, Neural networks for nonlinear and mixed complementarity problems and their applications, Nerual Networks, 17 (2004), 271-283. doi: 10.1016/j.neunet.2003.07.006. [4] Y. G. Evtushenko, Two numerical methods of solving nonlinear programming problems, Sov. Math. Dokl, 15 (1974), 420-423. [5] Y. G. Evtushenko, "Numerical Optimization Techniques," In: Optimization Software. New York: Inc. Publication Dvision, 1985. [6] F. Facchinei, A. Fischer and C. Kanzow, A semismooth Newton method for variational inequalities: The case of box constraint, Complementarity and Variational Problems (Baltimore, MD, 1995), SIAM, Philadelphia, PA, (1997), 76-90. . [7] F. Facchinei and J.-S. Pang, "Finite-dimensional Variational Inequalities and Complementarity Problems," volume II, Springer-Verlag New York, Inc., 2003. [8] A. V. Fiacco and G. P. Mccormick, "Nonlinear Programming: Sequential Unconstrained Minimization Techniques," John Wiley and Sons, Inc., New York-London-Sydney, 1968. [9] M. Fukushima, Equivalent differentiable optimization problems and descent method for asymmetric variatioanl inequality problems, Math. Program., 53 (1992), 99-110. doi: 10.1007/BF01585696. [10] T. L. Friesz, D. H. Bernstein, N. J. Mehta, R. L. Tobin and S. Ganjlizadeh, Day-to-day dynamic network disequilibria and idealized traveler information systems, Operations Research, 42 (1994), 1120-1136. doi: 10.1287/opre.42.6.1120. [11] X. B. Gao, Exponential stability of globally projected dynamic systems, IEEE Trans. Neural Networks, 14 (2003), 426-431. doi: 10.1109/TNN.2003.809409. [12] X. B. Gao, L. Liao and L. Qi, A novel neural network for variational inequalities with linear and nonlinear constraints, IEEE Transactions on Neural Networks, 16 (2005), 1305-1317. doi: 10.1109/TNN.2005.852974. [13] X. L. Hu and J. Wang, Solving pseudomonotone variational inequalities and pseu- doconvex optimization problems using the projection neural network, IEEE Trans. Neu-ral Networks, 17 (2006), 1487-1499. doi: 10.1109/TNN.2006.879774. [14] R. Horn and C. Johnson, "Matrix Analysis," Cambridge University Press, Cambridge, 1985. [15] L. Liao, H. Qi and L. Qi, Solving nonlinear complementarity problems with neural networks: a reformulation method approach, Journal of Computational and Applied Mathematics, 131 (2001), 343-359. doi: 10.1016/S0377-0427(00)00262-4. [16] U. Mosco, Implicit variational problems and quasi-variational inequalities, Lecture Note in Math., Springer-Verlag, Berlin, 543 (1976), 83-156. [17] L. Qi and J. Sun, A nonsmooth verson of Newton's method, Mathematical Programming, 58 (1993), 353-367. doi: 10.1007/BF01581275. [18] L. Qi, D. F. Sun and G. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Mathematical Programming, Ser. A, 87 (2000), 1-35. [19] D. F. Sun, A class of iterative methods for solving nonlinear projection equations, Optimization Theory and Applications, 91 (1996), 123-140. doi: 10.1007/BF02192286. [20] D. F. Sun and R. S. Womersley, A new unconstrained differentialble merit function for box constrained variational inequality problems and a damped Gauss-Newton method, SIAM J. Optim., 9 (1999), 388-413. doi: 10.1137/S1052623496314173. [21] D. F. Sun, The strong second order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications, Math. Oper. Res., 31 (2006), 761-776. doi: 10.1287/moor.1060.0195. [22] G. V. Smirnov, "Introduction to the Theory of Differential Inclusions," Graduates Studies in Mathematics, 41, American Mathematical Society, 2002. [23] Y. S. Xia and J. Wang, On the stability of globally projected dynamical systems, J. Optim. Theory Appl., 106 (2000), 129-150. doi: 10.1023/A:1004611224835. [24] Y. S. Xia, Further results on global convergence and stability of globally projected dynamic systems, Journal of Optim. Theory Appl., 122 (2004), 627-649. doi: 10.1023/B:JOTA.0000042598.21226.af. [25] J. Zabczyk, "Mathematical Control Theory: An Introduction," Birkhauser Boston Inc., Boston, 1992.

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##### References:
 [1] K. J. Arrow and L. Hurwicz, Reduction of constrained maxima to saddle point problems, in "Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability" (J. Neyman Ed.), University of California Press, Berkeleyand Los Angeles, 5 (1956), 1-20. [2] J. Chen, C. Ko and S. Pan, A neural network based on the generalized Fischer-Burmeister function for nonlinear complementarity problems, Information Sciences, 180 (1992), 697-711. doi: 10.1016/j.ins.2009.11.014. [3] C. Dang, Y. Leung, X. Gao and K. Chen, Neural networks for nonlinear and mixed complementarity problems and their applications, Nerual Networks, 17 (2004), 271-283. doi: 10.1016/j.neunet.2003.07.006. [4] Y. G. Evtushenko, Two numerical methods of solving nonlinear programming problems, Sov. Math. Dokl, 15 (1974), 420-423. [5] Y. G. Evtushenko, "Numerical Optimization Techniques," In: Optimization Software. New York: Inc. Publication Dvision, 1985. [6] F. Facchinei, A. Fischer and C. Kanzow, A semismooth Newton method for variational inequalities: The case of box constraint, Complementarity and Variational Problems (Baltimore, MD, 1995), SIAM, Philadelphia, PA, (1997), 76-90. . [7] F. Facchinei and J.-S. Pang, "Finite-dimensional Variational Inequalities and Complementarity Problems," volume II, Springer-Verlag New York, Inc., 2003. [8] A. V. Fiacco and G. P. Mccormick, "Nonlinear Programming: Sequential Unconstrained Minimization Techniques," John Wiley and Sons, Inc., New York-London-Sydney, 1968. [9] M. Fukushima, Equivalent differentiable optimization problems and descent method for asymmetric variatioanl inequality problems, Math. Program., 53 (1992), 99-110. doi: 10.1007/BF01585696. [10] T. L. Friesz, D. H. Bernstein, N. J. Mehta, R. L. Tobin and S. Ganjlizadeh, Day-to-day dynamic network disequilibria and idealized traveler information systems, Operations Research, 42 (1994), 1120-1136. doi: 10.1287/opre.42.6.1120. [11] X. B. Gao, Exponential stability of globally projected dynamic systems, IEEE Trans. Neural Networks, 14 (2003), 426-431. doi: 10.1109/TNN.2003.809409. [12] X. B. Gao, L. Liao and L. Qi, A novel neural network for variational inequalities with linear and nonlinear constraints, IEEE Transactions on Neural Networks, 16 (2005), 1305-1317. doi: 10.1109/TNN.2005.852974. [13] X. L. Hu and J. Wang, Solving pseudomonotone variational inequalities and pseu- doconvex optimization problems using the projection neural network, IEEE Trans. Neu-ral Networks, 17 (2006), 1487-1499. doi: 10.1109/TNN.2006.879774. [14] R. Horn and C. Johnson, "Matrix Analysis," Cambridge University Press, Cambridge, 1985. [15] L. Liao, H. Qi and L. Qi, Solving nonlinear complementarity problems with neural networks: a reformulation method approach, Journal of Computational and Applied Mathematics, 131 (2001), 343-359. doi: 10.1016/S0377-0427(00)00262-4. [16] U. Mosco, Implicit variational problems and quasi-variational inequalities, Lecture Note in Math., Springer-Verlag, Berlin, 543 (1976), 83-156. [17] L. Qi and J. Sun, A nonsmooth verson of Newton's method, Mathematical Programming, 58 (1993), 353-367. doi: 10.1007/BF01581275. [18] L. Qi, D. F. Sun and G. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Mathematical Programming, Ser. A, 87 (2000), 1-35. [19] D. F. Sun, A class of iterative methods for solving nonlinear projection equations, Optimization Theory and Applications, 91 (1996), 123-140. doi: 10.1007/BF02192286. [20] D. F. Sun and R. S. Womersley, A new unconstrained differentialble merit function for box constrained variational inequality problems and a damped Gauss-Newton method, SIAM J. Optim., 9 (1999), 388-413. doi: 10.1137/S1052623496314173. [21] D. F. Sun, The strong second order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications, Math. Oper. Res., 31 (2006), 761-776. doi: 10.1287/moor.1060.0195. [22] G. V. Smirnov, "Introduction to the Theory of Differential Inclusions," Graduates Studies in Mathematics, 41, American Mathematical Society, 2002. [23] Y. S. Xia and J. Wang, On the stability of globally projected dynamical systems, J. Optim. Theory Appl., 106 (2000), 129-150. doi: 10.1023/A:1004611224835. [24] Y. S. Xia, Further results on global convergence and stability of globally projected dynamic systems, Journal of Optim. Theory Appl., 122 (2004), 627-649. doi: 10.1023/B:JOTA.0000042598.21226.af. [25] J. Zabczyk, "Mathematical Control Theory: An Introduction," Birkhauser Boston Inc., Boston, 1992.
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