# American Institute of Mathematical Sciences

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. [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. [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. [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. [5] F. Facchinei and J. S. Pang, Finite-dimensional Variational Inequalities and Complementarity Problems,, Vol. II. Springer Series in Operations Research. Springer-Verlag, (2003). [6] J. Faraut and A. Korányi, Analysis on Symmetric Cones,, Oxford: Clarendon Press, (1994). [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. [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. [9] P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations,, Acta Mathematica, 115 (1966), 271. doi: 10.1007/BF02392210. [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. [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. [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. [13] J. Lions and G. Stampacchia, Variational inequalities,, Communications on Pure and Applied Mathematics, 20 (1967), 493. doi: 10.1002/cpa.3160200302. [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. [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. [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. [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. [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. [19] L. Qi and J. Sun, A nonsmooth version of Newton's method,, Mathematical programming, 58 (1993), 353. doi: 10.1007/BF01581275. [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. [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. [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.

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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. [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. [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. [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. [5] F. Facchinei and J. S. Pang, Finite-dimensional Variational Inequalities and Complementarity Problems,, Vol. II. Springer Series in Operations Research. Springer-Verlag, (2003). [6] J. Faraut and A. Korányi, Analysis on Symmetric Cones,, Oxford: Clarendon Press, (1994). [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. [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. [9] P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations,, Acta Mathematica, 115 (1966), 271. doi: 10.1007/BF02392210. [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. [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. [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. [13] J. Lions and G. Stampacchia, Variational inequalities,, Communications on Pure and Applied Mathematics, 20 (1967), 493. doi: 10.1002/cpa.3160200302. [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. [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. [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. [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. [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. [19] L. Qi and J. Sun, A nonsmooth version of Newton's method,, Mathematical programming, 58 (1993), 353. doi: 10.1007/BF01581275. [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. [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. [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.
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