Abstract
The functions involving in the nonlinear complementarity problems (NCP) are often defined only in nonnegative orthant. It is necessary to develop algorithms with the following properties: 1) all iterates generated remain in the nonnegative orthant; 2) any accumulation point of the iterates is a solution of NCP. In this paper, we reformulate the NCP as an equivalent constrained minimization problem with simple bounds. The reformulation is based on a class of functions, which generalize the squared Fischer-Burmeister NCP function. It is shown that the KKT point of the constrained minimization problem is a solution of NCP if the Jacobian matrix of the function involving in NCP isP 0-matrix, A global convergent method is proposed for monotone NCP with the desirable properties (1) and (2).
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References
B. Chen and P.T. Harker,A continuation approximations to nonlinear complementarity problems, SIAM Journal on Optimization, 7 (1997), 403–420
F. Facchinei and C. Kanzow,On unconstrained and constrained stationary points of implicit Lagrangian, Journal of Optimization Theory and Applications (1997).
F. Facchinei and J. Soares,A new merit function for nonlinear complementarity problems and a related algorithm, SIAM J. Optim. 7 (1997), PP. 225–247
M.C. Ferris and D. Ralph,Projected gradient methods for nonlinear complementarity problems via normal maps, in D.Z. Du, L. Qi and R.S. Womersley eds., Recent Advances in Nonsmooth Optimization, World Scientific Publishes, Singapore (1995).
A. Fischer,A new constrained optimization reformulation for complementarity problem, Tech. Report, Institute of Numerical Mathematics, Technical University of Dresden, Dresden, Germany, July 1995.
A. Fischer,An NCP-function and its use for the solution of complementarity problems, in D.Z. Du, L. Qi and R.S. Womersley eds., Recent Advances in Nonsmooth Optimization, World Scientific Publishes, (1995), pp. 261–289.
M. Fukushima,Merit functions for variational inequality and complementarity problems, G. Di Pillo and F. Giannessi (eds.): Nonlinear Optimization and Applications, Plenum Publishing, New York, 1996, PP. 155–170.
C. Geiger and C. Kanzow,On the resolution of monotone complementarity problems, Computational Optimization and Application, 5 (1996), pp. 155–173.
G. Isac,Complementarity Problems, Lecture notes in Mathematics, Springer-Verlag, New York, 1992.
C. Kanzow,Nonlinear complementarity as unconstrained optimization, Journal of Optimization Theory and Application, 88 (1996), pp. 139–155.
C. Kanzow and M. Fukushima,Equivalence of the generalized complementarity problem to differentiate unconstrained minimization, Journal of Optimization Theory ang Applications 91 (1996).
Z.Q. Luo and P. Tseng,A new class of merit functions for the nonlinear complementarity problem, In M.C. Ferris and J.S. Pang (eds.): Complementarity and Variational Problems: State of the Art. SIAM, Philadelphia, PA, 1997, pp. 204–225.
O.L. Mangasarian and M.V. Solodov,Nonlinear complementarity as unconstrained and constrained minimization, Mathematical Programming 62 B, pp.277–297.
J.J. Moré,Global methods for nonlinear complementarity problems, Preprint MCS-P429-0494, Argonne National Laboratory, Argonne, Illinois, April, 1994.
J.S. Pang,Complementarity problems, in R. Horst and P. pardalos eds., Handbook on Global Optimization, Klumer Academic Publishers, Norwell, Massachusetts, 1995., pp. 271–338.
J.S. Pang and S.A. Gabriel,NE/SQP: A robust algorithm for the nonlinear complementarity problem, Mathematical Programming 60, pp.295–337, 1993.
J.M. Peng,Unconstrained optimization methods for nonlinear complementarity problem, Journal of Computational Mathematics, Vol. 13, No. 3, 1995,pp. 259–266
J.M. Peng,A globally convergent method for monotone variational inequality problem with inequality constraints, Submitted to J. Optim. Theory Appl
J.M. Peng,Derivative-free methods for monotone variational inequality and complementarity problems, Institute of Computational Mathematics and Scientific/Engineering Computing, Academia Sinica, Beijing, China, Dec., 1996.
H.D. Qi,On minimizing and stationary sequences of a new class of merit functions for nonlinear complementarity problems, Institute of Computational Mathematics and Scientific/Engineering Computing, Academia Sinica, Beijing, China, Nov., 1996.
H.D. Qi and J.M. Peng,A new unconstrained optimization approach to nonlinear complementarity problems, Technical Report, Institute of Computational Mathematics and Scientific/Engineering Computing, Academia Sinica, Beijing, China, July, 1996.
H.D. Qi,A globally derivative-free method for nonlinear complementarity problems, Technical Report, Institute of Computational Mathematics and Scientific/Engineering Computing, Academia Sinica, Beijing, China, July, 1996.
P. Tseng,Growth behavior of a class of merit functions for the nonlinear complementarity problem, J. Optim. Theory Appl. 89 (1996), pp. 17–38.
P. Tseng, N. Yamashita and M. Fukushima,Equivalence of complementarity problems to differential minimization: A unified approach, SIAM Journal of Optimization 6 (1996), pp. 446–460.
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© 1998 Kluwer Academic Publishers
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Qi, HD. (1998). Algorithms Guaranteeing Iterative Points within Nonnegative Orthant in Complementarity Problems. In: Yuan, Yx. (eds) Advances in Nonlinear Programming. Applied Optimization, vol 14. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3335-7_16
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DOI: https://doi.org/10.1007/978-1-4613-3335-7_16
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-3337-1
Online ISBN: 978-1-4613-3335-7
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