Taiwanese Journal of Mathematics


Xiaona Fan, Tingting Xu, and Furong Gao

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In this paper, a smoothing homotopy method for solving the nonlinear complementarity problem is considered. The homotopy equation is constructed based on Chen-Harker-Kanzow-Smale smooth function. Under certain mild nonmonotone condition, the global convergence result is obtained. Furthermore, the initial point can be chosen almost everywhere in $R^n$ but not just in $R^n_+$. The numerical experimental results show that the method is effective.

Article information

Taiwanese J. Math., Volume 19, Number 1 (2015), 51-63.

First available in Project Euclid: 4 July 2017

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Zentralblatt MATH identifier

Primary: 90C33: Complementarity and equilibrium problems and variational inequalities (finite dimensions) 90C30: Nonlinear programming 65C20: Models, numerical methods [See also 68U20] 65L05: Initial value problems

nonlinear complementarity problem homotopy method smoothing method global convergence


Fan, Xiaona; Xu, Tingting; Gao, Furong. SOLVING NONLINEAR COMPLEMENTARITY PROBLEM BY A SMOOTHING HOMOTOPY METHOD. Taiwanese J. Math. 19 (2015), no. 1, 51--63. doi:10.11650/tjm.19.2015.3357. https://projecteuclid.org/euclid.twjm/1499133616

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  • E. L. Allgower and K. Georg, Numerical Continuation Methods: An Introduction, Springer-Vergal, Berlin, New York, 1990.
  • B. T. Chen and P. T. Harker, Smooth approximations to nonlinear complementarity problems, SIAM Journal on Optimization, \bf7(2) (1997), 403-420.
  • B. L. Chen and C. F. Ma, A new smoothing Broyden-like method for solving nonlinear complementarity problem with a $P_0$ function, J. Glob. Optim., \bf51 (2011), 473-495.
  • M. C. Ferris and J.-S. Pang, Engineering and economic applications of complementarity problems, SIAM Review, \bf39 (1997), 669-713.
  • M. Fukushima, Merit functions for variational inequality and complementarity problems, in Nonlinear Optimization and Applications, G. Di Pillo and F. Giannessi eds., Plenum Publishing Corporation, New York, pp. 155-170, 1996.
  • P. T. Harker and J.-S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problem: a survey of theory, algorithms and applications, Mathematical Progamming, \bf48 (1990), 161-220.
  • P. T. Harker, Complementarity problem, in Handbook of Global Optimization, R. Horst and P. Pardalos, eds., Kluwer Academic Publishers, Boston, pp. 271-338, 1995.
  • H. Jiang and L. Qi, A new nonsmooth equations approach to nonliner complementarity problems, SIAM Journal on Control and Optimization, \bf35 (1997), 178-193.
  • C. Kanzow, Some equation-based methods for the nonlinear complementarity problem, Optimization Methods and Software, \bf3 (1994), 327-340.
  • M. Kojima and S. Shindo, Extensions of Newton and quasi-Newton methods to systems of $PC^1$ equations, Journal of Operation Research Society of Japan, \bf29 (1986), 352-374.
  • T. De Luca, F. Facchinei and C. Kanzow, A semismooth equation approach to the solution of nonlinear complementarity problems, Mathematical Programming, \bf75(3) (1996), 407-439.
  • L. Mathiesen, An algorithm based on a sequence of linear complementarity problems applied to a Walrasian equilibrium model: an example, Mathematical Programming, \bf37 (1987), 1-18.
  • G. L. Naber, Topological Method in Euclidean Space, Cambridge Univ. Press, London, 1980.
  • D. Sun, A regularization Newton method for solving nonlinear complementarity problems, Applied Mathematics and Optimization, 40 (1999), 315-339.
  • L. T. Watson, Solving the nonlinear complementarity problem by a homotopy method, SIAM Journal on Control and Optimization, 17(1) (1979), 36-46.
  • Q. Xu and C. Dang, A new homotopy method for solving non-linear complementarity problems, Optimization, 57(5) (2008), 681-689.
  • B. Yu and Z. Lin, Homotopy method for a class of nonconvex Brouwer fixed point problems, Applied Mathematics and Computation, \bf74 (1996), 65-77.