# American Institute of Mathematical Sciences

April  2011, 7(2): 467-482. doi: 10.3934/jimo.2011.7.467

## A nonmonotone smoothing Newton algorithm for solving box constrained variational inequalities with a $P_0$ function

 1 College of Science, Civil Aviation University of China, Tianjin 300300, China 2 Department of Mathematics, School of Science, Tianjin University, Tianjin 300072

Received  March 2010 Revised  February 2011 Published  April 2011

In this paper, we propose a new class of smoothing functions which uniformly approximates the median function of three scalars. The proposed functions are the generalization of the smoothing function proposed by Li and Fukushima. Some favorable properties of the functions are investigated. By using the proposed functions, we reformulate the box constrained variational inequality problem (VIP) as a system of parameterized smooth equations, and then propose a smoothing Newton algorithm with a nonmonotone line search to solve the VIP. The proposed algorithm is proved to be globally and locally superlinearly convergent under suitable assumptions. Some numerical results for test problems from MCPLIB are also reported, which demonstrate that the proposed smoothing functions are valuable and the proposed algorithm is effective.
Citation: Na Zhao, Zheng-Hai Huang. A nonmonotone smoothing Newton algorithm for solving box constrained variational inequalities with a $P_0$ function. Journal of Industrial & Management Optimization, 2011, 7 (2) : 467-482. doi: 10.3934/jimo.2011.7.467
##### References:
 [1] S. C. Billups, S. P. Dirkse and M. C. Soares, A comparison of algorithms for large scale mixed complementarity problems,, Comput. Optim. Appl., 7 (1997), 3. doi: 10.1023/A:1008632215341. Google Scholar [2] B. Chen and X. Chen, A global linear and local quadratic continuation smoothing method for variational inequalities with box constraints,, Comput. Optim. Appl., 13 (2000), 131. doi: 10.1023/A:1026546230851. Google Scholar [3] C. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems,, Comput. Optim. Appl., 5 (1996), 97. doi: 10.1007/BF00249052. Google Scholar [4] 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. doi: 10.1007/s10898-006-9027-y. Google Scholar [5] J.-S. Chen and P. H. Pan, A family of NCP functions and a descent method for the nonlinear complementarity problem,, Comput. Optim. Appl., 40 (2008), 389. doi: 10.1007/s10589-007-9086-0. Google Scholar [6] J.-S. Chen, S. H. Pan and T. C. Lin, A smoothing Newton method based on the generalized Fischer-Burmeister function for MCPs,, Nonlinear Anal.: TMA, 72 (2010), 3739. doi: 10.1016/j.na.2010.01.012. Google Scholar [7] J.-S. Chen, S. H. Pan and C. Y. Yang, Numerical comparisons of two effective method for mixed complementarity problems,, J. Comput. Appl. Math., 234 (2010), 667. doi: 10.1016/j.cam.2010.01.004. Google Scholar [8] X. Chen, L. Qi and D. Sun, Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities,, Math. Comput., 67 (1998), 519. doi: 10.1090/S0025-5718-98-00932-6. Google Scholar [9] X. Chen and Y. Ye, On homotopy-smoothing methods for box-constrained variational inequalities,, SIAM J. Control Optim., 37 (1999), 589. doi: 10.1137/S0363012997315907. Google Scholar [10] F. Facchinei, A. Fischer and C. Kanzow, A semismooth Newton method for variational inequalities: The case of box constraints,, in, (1997), 76. Google Scholar [11] M. Ferris, C. Kanzow and T. S. Munson, Feasible descent algorithms for mixed complentarity problems,, Math. Program., 86 (1999), 475. doi: 10.1007/s101070050101. Google Scholar [12] A. Fischer, Solution of monotone complementarity problems with Lipschitzian functions,, Math. Program., 76 (1997), 513. doi: 10.1007/BF02614396. Google Scholar [13] M. S. Gowda and J. J. Sznajder, Weak univalence and connectedness of inverse images of continuous functions,, Math. Oper. Res., 24 (1999), 255. doi: 10.1287/moor.24.1.255. Google Scholar [14] M. S. Gowda and M. A. Tawhid, Existence and limiting behavior of trajectories associated with $P_0$-equations,, Comput. Optim. Appl., 12 (1999), 229. doi: 10.1023/A:1008688302346. Google Scholar [15] S. L. Hu and Z. H. Huang, Smoothness of a class of generalized merit functions for the second-order cone complementarity problem,, Pacific J. Optim., 6 (2010), 551. Google Scholar [16] S. L. Hu, Z. 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. doi: 10.1016/j.cam.2008.10.056. Google Scholar [17] S. L. Hu, Z. H. Huang and N. Lu, A non-monotone line search algorithm for unconstrained optimization,, J. Sci. Comput., 42 (2010), 38. doi: 10.1007/s10915-009-9314-0. Google Scholar [18] S. L. Hu, Z.H. Huang and P. Wang, A nonmonotone smoothing Newton algorithm for solving nonlinear complementarity problems,, Optim. Methods Softw., 24 (2009), 447. doi: 10.1080/10556780902769862. Google Scholar [19] Z. H. Huang, L. Qi and D. Sun, Sub-quadratic convergence of a smoothing Newton algorithm for the $P_0$- and monotone LCP,, Math. Program., 99 (2004), 423. doi: 10.1007/s10107-003-0457-8. Google Scholar [20] C. Kanzow and M. Fukushima, Theoretical and numerical investigation of the D-gap function for box constrained variational inequalities,, Math. Program., 83 (1998), 55. doi: 10.1007/BF02680550. Google Scholar [21] C. Kanzow and S. Petra, Projected filter trust region methods for a semismooth least squares formulation of mixed complementarity problems,, Optim. Methods Softw., 22 (2007), 713. doi: 10.1080/10556780701296455. Google Scholar [22] D. Li and M. Fukushima, Smoothing Newton and quasi-Newton methods for mixed complementarity problems,, Comput. Optim. Appl., 17 (2000), 203. doi: 10.1023/A:1026502415830. Google Scholar [23] J. M. Peng, C. Kanzow and M. Fukushima, A hybrid Josephy-Newton method for solving box constrained variational inequality problems via the D-gap function,, Optim. Methods Softw., 10 (1999), 687. doi: 10.1080/10556789908805734. Google Scholar [24] H. D. Qi, A regularized smoothing Newton method for box constrained variational inequality problems with $P_0$-functions,, SIAM J. Optim., 10 (2000), 315. doi: 10.1137/S1052623497324047. Google Scholar [25] L. Qi, D. Sun and G. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequality problems,, Math. Program., 87 (2000), 1. Google Scholar [26] L. Qi and J. Sun, A nonsmooth version of Newton's method,, Math. Program., 58 (1993), 353. doi: 10.1007/BF01581275. Google Scholar [27] G. Ravindran and M. S. Gowda, Regularization of $P_0$-functions in box variational inequality problems,, SIAM J. Optim., 11 (2001), 748. doi: 10.1137/S1052623497329567. Google Scholar [28] D. Sun and R. S. Womersley, A new unconstrained differentiable merit function for box constrained variational inequality problems and a damped Guass-Newton method,, SIAM J. Optim., 9 (1999), 388. doi: 10.1137/S1052623496314173. Google Scholar

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##### References:
 [1] S. C. Billups, S. P. Dirkse and M. C. Soares, A comparison of algorithms for large scale mixed complementarity problems,, Comput. Optim. Appl., 7 (1997), 3. doi: 10.1023/A:1008632215341. Google Scholar [2] B. Chen and X. Chen, A global linear and local quadratic continuation smoothing method for variational inequalities with box constraints,, Comput. Optim. Appl., 13 (2000), 131. doi: 10.1023/A:1026546230851. Google Scholar [3] C. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems,, Comput. Optim. Appl., 5 (1996), 97. doi: 10.1007/BF00249052. Google Scholar [4] 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. doi: 10.1007/s10898-006-9027-y. Google Scholar [5] J.-S. Chen and P. H. Pan, A family of NCP functions and a descent method for the nonlinear complementarity problem,, Comput. Optim. Appl., 40 (2008), 389. doi: 10.1007/s10589-007-9086-0. Google Scholar [6] J.-S. Chen, S. H. Pan and T. C. Lin, A smoothing Newton method based on the generalized Fischer-Burmeister function for MCPs,, Nonlinear Anal.: TMA, 72 (2010), 3739. doi: 10.1016/j.na.2010.01.012. Google Scholar [7] J.-S. Chen, S. H. Pan and C. Y. Yang, Numerical comparisons of two effective method for mixed complementarity problems,, J. Comput. Appl. Math., 234 (2010), 667. doi: 10.1016/j.cam.2010.01.004. Google Scholar [8] X. Chen, L. Qi and D. Sun, Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities,, Math. Comput., 67 (1998), 519. doi: 10.1090/S0025-5718-98-00932-6. Google Scholar [9] X. Chen and Y. Ye, On homotopy-smoothing methods for box-constrained variational inequalities,, SIAM J. Control Optim., 37 (1999), 589. doi: 10.1137/S0363012997315907. Google Scholar [10] F. Facchinei, A. Fischer and C. Kanzow, A semismooth Newton method for variational inequalities: The case of box constraints,, in, (1997), 76. Google Scholar [11] M. Ferris, C. Kanzow and T. S. Munson, Feasible descent algorithms for mixed complentarity problems,, Math. Program., 86 (1999), 475. doi: 10.1007/s101070050101. Google Scholar [12] A. Fischer, Solution of monotone complementarity problems with Lipschitzian functions,, Math. Program., 76 (1997), 513. doi: 10.1007/BF02614396. Google Scholar [13] M. S. Gowda and J. J. Sznajder, Weak univalence and connectedness of inverse images of continuous functions,, Math. Oper. Res., 24 (1999), 255. doi: 10.1287/moor.24.1.255. Google Scholar [14] M. S. Gowda and M. A. Tawhid, Existence and limiting behavior of trajectories associated with $P_0$-equations,, Comput. Optim. Appl., 12 (1999), 229. doi: 10.1023/A:1008688302346. Google Scholar [15] S. L. Hu and Z. H. Huang, Smoothness of a class of generalized merit functions for the second-order cone complementarity problem,, Pacific J. Optim., 6 (2010), 551. Google Scholar [16] S. L. Hu, Z. 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. doi: 10.1016/j.cam.2008.10.056. Google Scholar [17] S. L. Hu, Z. H. Huang and N. Lu, A non-monotone line search algorithm for unconstrained optimization,, J. Sci. Comput., 42 (2010), 38. doi: 10.1007/s10915-009-9314-0. Google Scholar [18] S. L. Hu, Z.H. Huang and P. Wang, A nonmonotone smoothing Newton algorithm for solving nonlinear complementarity problems,, Optim. Methods Softw., 24 (2009), 447. doi: 10.1080/10556780902769862. Google Scholar [19] Z. H. Huang, L. Qi and D. Sun, Sub-quadratic convergence of a smoothing Newton algorithm for the $P_0$- and monotone LCP,, Math. Program., 99 (2004), 423. doi: 10.1007/s10107-003-0457-8. Google Scholar [20] C. Kanzow and M. Fukushima, Theoretical and numerical investigation of the D-gap function for box constrained variational inequalities,, Math. Program., 83 (1998), 55. doi: 10.1007/BF02680550. Google Scholar [21] C. Kanzow and S. Petra, Projected filter trust region methods for a semismooth least squares formulation of mixed complementarity problems,, Optim. Methods Softw., 22 (2007), 713. doi: 10.1080/10556780701296455. Google Scholar [22] D. Li and M. Fukushima, Smoothing Newton and quasi-Newton methods for mixed complementarity problems,, Comput. Optim. Appl., 17 (2000), 203. doi: 10.1023/A:1026502415830. Google Scholar [23] J. M. Peng, C. Kanzow and M. Fukushima, A hybrid Josephy-Newton method for solving box constrained variational inequality problems via the D-gap function,, Optim. Methods Softw., 10 (1999), 687. doi: 10.1080/10556789908805734. Google Scholar [24] H. D. Qi, A regularized smoothing Newton method for box constrained variational inequality problems with $P_0$-functions,, SIAM J. Optim., 10 (2000), 315. doi: 10.1137/S1052623497324047. Google Scholar [25] L. Qi, D. Sun and G. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequality problems,, Math. Program., 87 (2000), 1. Google Scholar [26] L. Qi and J. Sun, A nonsmooth version of Newton's method,, Math. Program., 58 (1993), 353. doi: 10.1007/BF01581275. Google Scholar [27] G. Ravindran and M. S. Gowda, Regularization of $P_0$-functions in box variational inequality problems,, SIAM J. Optim., 11 (2001), 748. doi: 10.1137/S1052623497329567. Google Scholar [28] D. Sun and R. S. Womersley, A new unconstrained differentiable merit function for box constrained variational inequality problems and a damped Guass-Newton method,, SIAM J. Optim., 9 (1999), 388. doi: 10.1137/S1052623496314173. Google Scholar
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