A nonmonotone inexact Broyden-like algorithm for nonlinear complementarity problems

Abstract

In this paper, by constructing a new smoothing complementary function, we reformulate the 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.

Keywords

Complementarity problem / global convergence / local quadratic convergence / Broyden-like algorithm

References

[1] Chen B.L. , 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. , Xie, Y.J.. A Nonmonotone Broyden-like method for nonlinear complementarity problems. Math. Numer. Sin. 2013; 36(2): 181–194

[3] Harker P.T. , 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. , Chen, Z.W.. Predictor-corrector smoothing Newton method, based on a new smoothing function, for solving the nonlinear complementarity problem with a Po function. J. Optim. Theory Appl. 2003; 117: 39–68

[5] Jiang H.Y. , Qi, L.Q.. A new nonsmooth equations approach to nonlinear complementarity problems. SIAM J. Control Optim. 1997; 35: 178–193

[6] Liu R.J. , 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. , Wang, D.Q.. Study on nonlinear complementarity problem by using the Fischer-Burmeister function. J. Inner Mongolia Agricultural Univ. 2006; 27(2): 133–134

[8] Wang Y.J., Ma, F.M. , 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. , Zhang, L.. An inexact Broyden method for nonlinear equations. Math. Theory Appl. 2016; 36(2): 1–9

[10] Zheng X.Y., Shi, J.R., Yang, W. , Yin, Q.Y.. Nonmonotone smoothing Broyden-like method for generalized nonlinear complementarity problems. J. Appl. Math. Comput. 2017; 54: 277–295

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Higher Education Press 2025