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Smoothing inexact Newton method based on a new derivative-free nonmonotone line search for the NCP over circular cones

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Abstract

In this paper we consider the nonlinear complementarity problem over circular cones (CCNCP) which contains a lot of circular cone optimization problems. We study a one-parametric class of smoothing functions which can be used to reformulate the CCNCP as a system of smooth nonlinear equations. Based on the equivalent reformulation, we propose a smoothing inexact Newton method to solve the CCNCP. In each iteration, the proposed method solves the nonlinear equations only approximately. Since the inexact direction is not necessarily descent, a new derivative-free nonmonotone line search is developed to ensure that the proposed method has global and local superlinear and quadratical convergence. Some numerical results are also reported.

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References

  • Alzalg, B. (2017). The Jordan algebraic structure of the circular cone. Operators and Matrices, 11(1), 1–21.

    Google Scholar 

  • Bai, Y. Q., Gao, X. R., & Wang, G. Q. (2015). Primal-dual interior-point algorithms for convex quadratic circular cone optimization. Numerical Algebra, Control and Optimization, 5, 211–231.

    Google Scholar 

  • Bai, Y. Q., Ma, P. F., & Zhang, J. (2016). A polynomial-time interior-point method for circular cone programming based on kernel functions. Journal of Industrial and Management Optimization, 12, 739–756.

    Google Scholar 

  • Chen, J. S., & Pan, S. H. (2010). A one-parametric class of merit functions for the second-order cone complementarity problem. Computational Optimization and Applications, 45, 581–606.

    Google Scholar 

  • Chen, J. S., & Pan, S. H. (2012). A survey on SOC complementarity functions and solution methods for SOCPs and SOCCPs. Pacific Journal of Optimization, 8(1), 33–74.

    Google Scholar 

  • Chi, X. N., Wan, Z. P., Zhu, Z. B., & Yuan, L. Y. (2016). A nonmonotone smoothing Newton method for circular cone programming. Optimization, 65(12), 2227–2250.

    Google Scholar 

  • Chi, X. N., Wei, H. J., Wan, Z. P., & Zhu, Z. B. (2017). Smoothing Newton algorithm for the circular cone programming with a nonmonotone line search. Acta Mathematica Scientia, 37B(5), 1262–1280.

    Google Scholar 

  • Chi, X. N., Wei, H. J., Wan, Z. P., & Zhu, Z. B. (2019). A nonmonotone smoothing Newton algorithm for circular cone complementarity problems. Journal of Computational Analysis and Applications, 26, 146–162.

    Google Scholar 

  • Dattorro, J. (2005). Convex optimization and Euclidean distance geometry. California: Meboo Publlishing.

    Google Scholar 

  • Dembo, R. S., Eisenstat, S. C., & Steihaug, T. (1982). Inexact Newton methods. SIAM Journal on Numerical Analysis, 19(2), 400–408.

    Google Scholar 

  • Dong, L., Tang, J. Y., & Song, X. Y. (2018). Numerical study of a smoothing algorithm for the complementarity system over the second-order cone. Computational and Applied Mathematics, 37, 2845–2861.

    Google Scholar 

  • Fukushima, M., Luo, Z., & Tseng, P. (2001). Smoothing functions for second-order-cone complementarity problems. SIAM Journal on Optimization, 12, 36–460.

    Google Scholar 

  • Facchinei, F., & Pang, J. S. (2003). Finite-dimensional variational inequalities and complementarity problems, vol. I and II. New York: Springer.

    Google Scholar 

  • Huang, Z. H., Hu, S. L., & Han, J. Y. (2009). Convergence of a smoothing algorithm for symmetric cone complementarity problems with a nonmonotone line search. Science in China, Series A: Mathematics, 52, 833–848.

    Google Scholar 

  • Huang, Z. H., & Ni, T. (2010). Smoothing algorithms for complementarity problems over symmetric cones. Computational Optimization and Applications, 45, 557–579.

    Google Scholar 

  • Huang, Z. H., & Liu, X. H. (2011). Extension of smoothing Newton algorithms for solve linear programming over symmetric cones. Journal of Systems Science and Complexity, 24, 195–206.

    Google Scholar 

  • Kanzow, C., Ferenczi, I., & Fukushima, M. (2009). On the local convergence of semismooth Newton methods for linear and nonlinear second-order cone programs without strict complementarity. SIAM Journal on Optimization, 20, 297–320.

    Google Scholar 

  • Ke, Y. F., Ma, C. F., & Zhang, H. (2018). The relaxation modulus-based matrix splitting iteration methods for circular cone nonlinear complementarity problems. Journal of Computational and Applied Mathematics, 37(5), 6795–6820.

    Google Scholar 

  • Kheirfam, B., & Wang, G. Q. (2017). An infeasible full NT-step interior point method for circular optimization. Numerical Algebra, Control and Optimization, 7(2), 171–184.

    Google Scholar 

  • Kong, L. (2012). Quadratic convergence of a smoothing Newton method for symmetric cone programming without strict complementarity. Positivity, 16, 297–319.

    Google Scholar 

  • Kong, L., Sun, J., & Xiu, N. (2008). A regularized smoothing Newton method for symmetric cone complementarity problems. SIAM Journal on Optimization, 19, 1028–1047.

    Google Scholar 

  • Li, D. H., & Fukushima, M. (2000). A derivative-free line search and global convergence of Broyden-like method for nonlinear equations. Optimization Methods and Software, 13(3), 181–201.

    Google Scholar 

  • Li, Y. M., & Wei, D. Y. (2015). A generalized smoothing Newton method for the symmetric cone complementarity problem. Applied Mathematics and Computation, 264, 335–345.

    Google Scholar 

  • Liu, X. H., & Huang, Z. H. (2009). A smoothing Newton algorithm based on a one-parametric class of smoothing functions for linear programming over symmetric cones. Mathematical Methods of Operations Research, 70, 385–404.

    Google Scholar 

  • Lu, N., & Huang, Z. H. (2014). A smoothing Newton algorithm for a class of non-monotonic symmetric cone linear complementarity problems. Journal of Optimization Theory and Applications, 161, 446–464.

    Google Scholar 

  • Ma, C. F. (2011). A regularized smoothing Newton method for solving the symmetric cone complementarity problem. Mathematical and Computer Modelling, 54, 2515–2527.

    Google Scholar 

  • Ma, P. F., Bai, Y. Q., & Chen, J.-S. (2016). A self-concordant interior point algorithm for nonsymmetric circular cone programming. Journal of Nonlinear and Convex Analysis, 17(2), 225–241.

    Google Scholar 

  • Miao, X. H., Guo, S. J., Qi, N., & Chen, J. S. (2016). Constructions of complementarity functions and merit functions for circular cone complementarity problem. Computational Optimization and Applications, 63, 495–522.

    Google Scholar 

  • Miao, X. H., Yang, J. T., & Hu, S. L. (2015). A generalized Newton method for absolute value equations associated with circular cones. Applied Mathematics and Computation, 269, 155–168.

    Google Scholar 

  • Narushima, Y., Sagara, N., & Ogasawara, H. (2011). A smoothing Newton method with Fischer-Burmeister function for second-order cone complementarity problems. Journal of Optimization Theory and Applications, 149, 79–101.

    Google Scholar 

  • Pirhaji, M., Zangiabadi, M., & Mansouri, H. (2018). A path following interior-point method for linear complementarity problems over circular cones. Japan Journal of Industrial and Applied Mathematics, 35(3), 1103–1121.

    Google Scholar 

  • Qi, L. (1993). Convergence analysis of some algorithms for solving nonsmooth equations. Mathematical Operations Research, 18(1), 227–244.

    Google Scholar 

  • Qi, L., & Sun, J. (1993). A nonsmooth version of Newton’s method. Mathematical Programming, 58, 353–367.

    Google Scholar 

  • Sun, D., & Sun, J. (2005). Strong semismoothness of Fischer–Burmeister SDC and SOC complementarity functions. Mathematical Programming, 103, 575–581.

    Google Scholar 

  • Tang, J., & Ma, C. F. (2015). A smoothing Newton method for symmetric cone complementarity problems. Optimization Letters, 9(2), 225–244.

    Google Scholar 

  • Tang, J. Y., Zhou, J. C., & Fang, L. (2018). Strong convergence properties of a modified nonmonotone smoothing algorithm for the SCCP. Optimization Letters, 12, 411–424.

    Google Scholar 

  • Yoshise, A. (2006). Interior point trajectories, a homogeneous model for nonlinear complementarity problems over symmetric cones. SIAM Journal on Optimization, 17, 1129–1153.

    Google Scholar 

  • Zhang, H. C., & Hager, W. W. (2004). A nonmonotone line search technique and its application to unconstrained optimization. SIAM Journal on Optimization, 14, 1043–1056.

    Google Scholar 

  • Zhou, J. C., & Chen, J. S. (2013). Properties of circular cone and spectral factorization associated with circular cone. Journal of Nonlinear and Convex Analysis, 14, 807–816.

    Google Scholar 

  • Zhou, J. C., Chen, J. S., & Hung, H. F. (2013). Circular cone convexity and some inequalities associated with circular cones. Journal of Inequalities and Applications, 2013, 1–17.

    Google Scholar 

  • Zhou, J. C., Chen, J. S., & Mordukhovich, B. S. (2014). Variational analysis of circular cone programs. Optimization, 64, 113–147.

    Google Scholar 

  • Zhou, J. C., Tang, J. Y., & Chen, J. S. (2017). Parabolic second-order directional differentiability in the Hadamard sense of the vector-valued functions associated with circular cones. Journal of Optimization Theory and Applications, 172, 802–823.

    Google Scholar 

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Acknowledgements

This work was supported by National Natural Science Foundation of China (11771255, 11801325), Young Innovation Teams of Shandong Province (2019KJI013), Program of Science and Technology Activities for Overseas Students in Henan Province in 2020 and Nanhu Scholars Program for Young Scholars of Xinyang Normal University. We are very grateful to the two referees for their valuable comments on the paper which have considerably improved the paper.

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  1. School of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, China

    Jingyong Tang

  2. Department of Mathematics, Shandong University of Technology, Zibo, 255049, China

    Jinchuan Zhou

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  1. Jingyong Tang
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  2. Jinchuan Zhou
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Correspondence to Jingyong Tang.

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Tang, J., Zhou, J. Smoothing inexact Newton method based on a new derivative-free nonmonotone line search for the NCP over circular cones. Ann Oper Res 295, 787–808 (2020). https://doi.org/10.1007/s10479-020-03773-8

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