解非线性互补问题的非单调非精确Broyden-like算法
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摘要
本文通过构造一个新的光滑互补函数,将非线性互补问题等价转换为光滑方程组问题.将非单调线搜索技术与非精确Broyden-like算法相结合,建立了解非线性互补问题的非单调非精确Broyden-like算法.在一定条件下证明了该算法的全局收敛性和局部二次收敛性.数值实验表明该算法对求解非线性互补问题是十分有效的.
In this paper, by constructing a new smoothing complementary function, we reformulate nonlinear complementarity problem as a nonlinear smooth system of equations. Combining non-monotonic line search techniques with an inexact Broyden-like algorithm, we establish a nonmonotone inexact Broyden-like algorithm. The global and local quadratic convergence of this method is proved under suitable conditions. Numerical experiments show that the algorithm is effective for solving nonlinear complementarity problems.
In this paper, by constructing a new smoothing complementary function, we reformulate nonlinear complementarity problem as a nonlinear smooth system of equations. Combining non-monotonic line search techniques with an inexact Broyden-like algorithm, we establish a nonmonotone inexact Broyden-like algorithm. The global and local quadratic convergence of this method is proved under suitable conditions. Numerical experiments show that the algorithm is effective for solving nonlinear complementarity problems.
引文
[1] Chen, B.L. and Ma, C.F., Superlinear/quadratic smoothing Broyden-like method for the generalized nonlinear complementarity problem, Nonlinear Anal. Real World Appl., 2011, 12:1250-1263.
[2] Fan, B., Ma, C.F. and Xie, Y.J., A Nonmonotone Broyden-like method for nonlinear complementarity problems, Math. Numer. Sin., 2013, 36(2):181-194(in Chinese).
[3] Harker, P.T. and Pang, J.S., Finite-dimensional variational inequality and nonlinear complementarity problems:a survey of theory, algorithms and applications, Math. Program., 1990, 48:161-220.
[4] Huang, Z.H., Han, J.Y. and Chen, Z.W., Predictor-corrector smoothing Newton method, based on a new smoothing function, for solving the nonlinear complementarity problem with a P_0 function, J. Optim. Theory Appl., 2003, 117:39-68.
[5] Jiang, H.Y. and Qi, L.Q., A new nonsmooth equations approach to nonlinear complementarity problems,SIAM J. Control Optim., 1997, 35:178-193.
[6] Liu, R.J. and Dong, L., Nonmonotone smoothing inexact Newton method for the nonlinear complementarity problem, J, Appl. Math. Comput., 2016, 51:659-674.
[7] Wang, D.G., Pan, X. and Wang, D.Q., Study on nonlinear complementarity problem by using the FischerBurmeister function, J. Inner Mongolia Agricultural Univ., 2006, 27(2):133-134(in Chinese).
[8] Wang, Y.J., Ma, F.M. and Zhang, J.Z., A nonsmooth L-M method for solving the generalized nonlinear complementarity problem over a polyhedral cone, Appl. Math. Optim., 2005, 52:73-92.
[9] Wu, P.Y. and Zhang, L., An inexact Broyden method for nonlinear equations, Math. Theory Appl., 2016,36(2):1-9(in Chinese).
[10] Zheng, X.Y., Shi, J.R., Yang, W. and Yin, Q.Y., Nonmonotone smoothing Broyden-like method for generalized nonlinear complementarity problems, J. Appl. Math. Comput., 2017, 54:277-295.
[2] Fan, B., Ma, C.F. and Xie, Y.J., A Nonmonotone Broyden-like method for nonlinear complementarity problems, Math. Numer. Sin., 2013, 36(2):181-194(in Chinese).
[3] Harker, P.T. and Pang, J.S., Finite-dimensional variational inequality and nonlinear complementarity problems:a survey of theory, algorithms and applications, Math. Program., 1990, 48:161-220.
[4] Huang, Z.H., Han, J.Y. and Chen, Z.W., Predictor-corrector smoothing Newton method, based on a new smoothing function, for solving the nonlinear complementarity problem with a P_0 function, J. Optim. Theory Appl., 2003, 117:39-68.
[5] Jiang, H.Y. and Qi, L.Q., A new nonsmooth equations approach to nonlinear complementarity problems,SIAM J. Control Optim., 1997, 35:178-193.
[6] Liu, R.J. and Dong, L., Nonmonotone smoothing inexact Newton method for the nonlinear complementarity problem, J, Appl. Math. Comput., 2016, 51:659-674.
[7] Wang, D.G., Pan, X. and Wang, D.Q., Study on nonlinear complementarity problem by using the FischerBurmeister function, J. Inner Mongolia Agricultural Univ., 2006, 27(2):133-134(in Chinese).
[8] Wang, Y.J., Ma, F.M. and Zhang, J.Z., A nonsmooth L-M method for solving the generalized nonlinear complementarity problem over a polyhedral cone, Appl. Math. Optim., 2005, 52:73-92.
[9] Wu, P.Y. and Zhang, L., An inexact Broyden method for nonlinear equations, Math. Theory Appl., 2016,36(2):1-9(in Chinese).
[10] Zheng, X.Y., Shi, J.R., Yang, W. and Yin, Q.Y., Nonmonotone smoothing Broyden-like method for generalized nonlinear complementarity problems, J. Appl. Math. Comput., 2017, 54:277-295.