# American Institute of Mathematical Sciences

July  2015, 11(3): 733-746. doi: 10.3934/jimo.2015.11.733

## An inexact semismooth Newton method for variational inequality with symmetric cone constraints

 1 School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, Liaoning, China 2 Information and Engineering College, Dalian University, Dalian 116622, China

Received  September 2013 Revised  June 2014 Published  October 2014

In this paper, we consider using the inexact nonsmooth Newton method to efficiently solve the symmetric cone constrained variational inequality (VISCC) problem. It red provides a unified framework for dealing with the variational inequality with nonlinear constraints, variational inequality with the second-order cone constraints, and the variational inequality with semidefinite cone constraints. We get convergence of the above method and apply the results to three special types symmetric cones.
Citation: Shuang Chen, Li-Ping Pang, Dan Li. An inexact semismooth Newton method for variational inequality with symmetric cone constraints. Journal of Industrial & Management Optimization, 2015, 11 (3) : 733-746. doi: 10.3934/jimo.2015.11.733
##### References:
 [1] S. Bi, S. Pan and J. S. Chen, The same growth of FB and NR symmetric cone complementarity functions,, Optimization Letters, 6 (2012), 153.  doi: 10.1007/s11590-010-0257-z.  Google Scholar [2] S. Chen, L. P. Pang, F. F. Guo and Z. Q. Xia, Stochastic methods based on Newton method to the stochastic variational inequality problem with constraint conditions,, Mathematical and Computer Modelling, 55 (2012), 779.  doi: 10.1016/j.mcm.2011.09.003.  Google Scholar [3] X. D. Chen, D. Sun and J. Sun, Complementarity functions and numerical experiments on some smoothing Newton methods for second-order-cone complementarity problems,, Computational Optimization and Applications, 25 (2003), 39.  doi: 10.1023/A:1022996819381.  Google Scholar [4] F. Facchinei, A. Fischer, C. Kanzow and J. M. Peng, A simply constrained optimization reformulation of KKT systems arising from variational inequalities,, Applied Mathematics and Optimization, 40 (1999), 19.  doi: 10.1007/s002459900114.  Google Scholar [5] F. Facchinei and J. S. Pang, Finite-dimensional Variational Inequalities and Complementarity Problems,, Vol. II. Springer Series in Operations Research. Springer-Verlag, (2003).   Google Scholar [6] J. Faraut and A. Korányi, Analysis on Symmetric Cones,, Oxford: Clarendon Press, (1994).   Google Scholar [7] M. Fukushima, Z. Q. Luo and P. Tseng, Smoothing functions for second-order-cone complementarity problems,, SIAM Journal on optimization, 12 (2002), 436.  doi: 10.1137/S1052623400380365.  Google Scholar [8] M. S. Gowda, R. Sznajder and J. Tao, Some P-properties for linear transformations on Euclidean Jordan algebras,, Linear Algebra and Its Applications, 393 (2004), 203.  doi: 10.1016/j.laa.2004.03.028.  Google Scholar [9] P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations,, Acta Mathematica, 115 (1966), 271.  doi: 10.1007/BF02392210.  Google Scholar [10] S. Hayashi, N. Yamashita and M. Fukushima, A combined smoothing and regularization method for monotone second-order cone complementarity problems,, SIAM Journal on Optimization, 15 (2005), 593.  doi: 10.1137/S1052623403421516.  Google Scholar [11] L. Kong and Q. Meng, A semismooth Newton method for nonlinear symmetric cone programming,, Mathematical Methods of Operations Research, 76 (2012), 129.  doi: 10.1007/s00186-012-0393-6.  Google Scholar [12] L. Kong, J. Sun and N. Xiu, A regularized smoothing Newton method for symmetric cone complementarity problems,, SIAM Journal on Optimization, 19 (2008), 1028.  doi: 10.1137/060676775.  Google Scholar [13] J. Lions and G. Stampacchia, Variational inequalities,, Communications on Pure and Applied Mathematics, 20 (1967), 493.  doi: 10.1002/cpa.3160200302.  Google Scholar [14] L. Liu, S. Liu and H. Liu, A predictor-corrector smoothing Newton method for symmetric cone complementarity problems,, Applied Mathematics and Computation, 217 (2010), 2989.  doi: 10.1016/j.amc.2010.08.032.  Google Scholar [15] O. G. Mancino and G. Stampacchia, Convex programming and variational inequalities,, Journal of Optimization Theory and Applications, 9 (1972), 3.  doi: 10.1007/BF00932801.  Google Scholar [16] S. Pan, Y. L. Chang and J. S. Chen, Stationary point conditions for the FB merit function associated with symmetric cones,, Operations Research Letters, 38 (2010), 372.  doi: 10.1016/j.orl.2010.07.011.  Google Scholar [17] S. Pan and J. S. Chen, A semismooth Newton method for SOCCPs based on a one-parametric class of SOC complementarity functions,, Computational Optimization and Applications, 45 (2010), 59.  doi: 10.1007/s10589-008-9166-9.  Google Scholar [18] L. Qi, Convergence analysis of some algorithms for solving nonsmooth equations,, Mathematics of operations research, 18 (1993), 227.  doi: 10.1287/moor.18.1.227.  Google Scholar [19] L. Qi and J. Sun, A nonsmooth version of Newton's method,, Mathematical programming, 58 (1993), 353.  doi: 10.1007/BF01581275.  Google Scholar [20] D. Sun and J. Sun, Löwner's operator and spectral functions in Euclidean Jordan algebras,, Mathematics of Operations Research, 33 (2008), 421.  doi: 10.1287/moor.1070.0300.  Google Scholar [21] J. Sun, J. S. Chen and C. H. Ko, Neural networks for solving second-order cone constrained variational inequality problem,, Computational Optimization and Applications, 51 (2012), 623.  doi: 10.1007/s10589-010-9359-x.  Google Scholar [22] J. Zhang and K. Zhang, An inexact smoothing method for the monotone complementarity problem over symmetric cones,, Optimization Methods and Software, 27 (2012), 445.  doi: 10.1080/10556788.2010.534164.  Google Scholar

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##### References:
 [1] S. Bi, S. Pan and J. S. Chen, The same growth of FB and NR symmetric cone complementarity functions,, Optimization Letters, 6 (2012), 153.  doi: 10.1007/s11590-010-0257-z.  Google Scholar [2] S. Chen, L. P. Pang, F. F. Guo and Z. Q. Xia, Stochastic methods based on Newton method to the stochastic variational inequality problem with constraint conditions,, Mathematical and Computer Modelling, 55 (2012), 779.  doi: 10.1016/j.mcm.2011.09.003.  Google Scholar [3] X. D. Chen, D. Sun and J. Sun, Complementarity functions and numerical experiments on some smoothing Newton methods for second-order-cone complementarity problems,, Computational Optimization and Applications, 25 (2003), 39.  doi: 10.1023/A:1022996819381.  Google Scholar [4] F. Facchinei, A. Fischer, C. Kanzow and J. M. Peng, A simply constrained optimization reformulation of KKT systems arising from variational inequalities,, Applied Mathematics and Optimization, 40 (1999), 19.  doi: 10.1007/s002459900114.  Google Scholar [5] F. Facchinei and J. S. Pang, Finite-dimensional Variational Inequalities and Complementarity Problems,, Vol. II. Springer Series in Operations Research. Springer-Verlag, (2003).   Google Scholar [6] J. Faraut and A. Korányi, Analysis on Symmetric Cones,, Oxford: Clarendon Press, (1994).   Google Scholar [7] M. Fukushima, Z. Q. Luo and P. Tseng, Smoothing functions for second-order-cone complementarity problems,, SIAM Journal on optimization, 12 (2002), 436.  doi: 10.1137/S1052623400380365.  Google Scholar [8] M. S. Gowda, R. Sznajder and J. Tao, Some P-properties for linear transformations on Euclidean Jordan algebras,, Linear Algebra and Its Applications, 393 (2004), 203.  doi: 10.1016/j.laa.2004.03.028.  Google Scholar [9] P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations,, Acta Mathematica, 115 (1966), 271.  doi: 10.1007/BF02392210.  Google Scholar [10] S. Hayashi, N. Yamashita and M. Fukushima, A combined smoothing and regularization method for monotone second-order cone complementarity problems,, SIAM Journal on Optimization, 15 (2005), 593.  doi: 10.1137/S1052623403421516.  Google Scholar [11] L. Kong and Q. Meng, A semismooth Newton method for nonlinear symmetric cone programming,, Mathematical Methods of Operations Research, 76 (2012), 129.  doi: 10.1007/s00186-012-0393-6.  Google Scholar [12] L. Kong, J. Sun and N. Xiu, A regularized smoothing Newton method for symmetric cone complementarity problems,, SIAM Journal on Optimization, 19 (2008), 1028.  doi: 10.1137/060676775.  Google Scholar [13] J. Lions and G. Stampacchia, Variational inequalities,, Communications on Pure and Applied Mathematics, 20 (1967), 493.  doi: 10.1002/cpa.3160200302.  Google Scholar [14] L. Liu, S. Liu and H. Liu, A predictor-corrector smoothing Newton method for symmetric cone complementarity problems,, Applied Mathematics and Computation, 217 (2010), 2989.  doi: 10.1016/j.amc.2010.08.032.  Google Scholar [15] O. G. Mancino and G. Stampacchia, Convex programming and variational inequalities,, Journal of Optimization Theory and Applications, 9 (1972), 3.  doi: 10.1007/BF00932801.  Google Scholar [16] S. Pan, Y. L. Chang and J. S. Chen, Stationary point conditions for the FB merit function associated with symmetric cones,, Operations Research Letters, 38 (2010), 372.  doi: 10.1016/j.orl.2010.07.011.  Google Scholar [17] S. Pan and J. S. Chen, A semismooth Newton method for SOCCPs based on a one-parametric class of SOC complementarity functions,, Computational Optimization and Applications, 45 (2010), 59.  doi: 10.1007/s10589-008-9166-9.  Google Scholar [18] L. Qi, Convergence analysis of some algorithms for solving nonsmooth equations,, Mathematics of operations research, 18 (1993), 227.  doi: 10.1287/moor.18.1.227.  Google Scholar [19] L. Qi and J. Sun, A nonsmooth version of Newton's method,, Mathematical programming, 58 (1993), 353.  doi: 10.1007/BF01581275.  Google Scholar [20] D. Sun and J. Sun, Löwner's operator and spectral functions in Euclidean Jordan algebras,, Mathematics of Operations Research, 33 (2008), 421.  doi: 10.1287/moor.1070.0300.  Google Scholar [21] J. Sun, J. S. Chen and C. H. Ko, Neural networks for solving second-order cone constrained variational inequality problem,, Computational Optimization and Applications, 51 (2012), 623.  doi: 10.1007/s10589-010-9359-x.  Google Scholar [22] J. Zhang and K. Zhang, An inexact smoothing method for the monotone complementarity problem over symmetric cones,, Optimization Methods and Software, 27 (2012), 445.  doi: 10.1080/10556788.2010.534164.  Google Scholar
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