Convergence and error bound of a method for solving variational inequality problems via the generalized D-gap function. (English) Zbl 1045.49015

Qu, B.; Wang, C. Y.; Zhang, J. Z.

J. Optimization Theory Appl. 119, No. 3, 535-552 (2003).

Summary: The Variational Inequality Problem (VIP) can be reformulated as an unconstrained minimization problem through the generalized D-gap function. Recently, a hybrid Newton-type method was proposed by J. M. Peng and M. Fukushima [J. Math. Program. 86 A, No. 2, 367–386 (1999; Zbl 0939.90023)] for minimizing a special form of the generalized D-gap function. In this paper, the hybrid Newton-type algorithm is extended to minimize the general form \(g_{\alpha\beta}\) of the generalized D-gap function. It is shown that the algorithm has nice convergence properties. Under some reasonable conditions, it is proved that the algorithm is locally and globally convergent. Moreover, it is proved that the function \(g_{\alpha\beta}\) has bounded level sets for strongly monotone VIP. An error bound of the algorithm is obtained.

MSC:

49J40	Variational inequalities
90C33	Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
47J25	Iterative procedures involving nonlinear operators

Keywords:

variational inequality problems; unconstrained optimization; generalized D-gap function; global convergence; error bound estimation

Citations:

Zbl 0939.90023

References:

[1]	Harker, P. T., and Pang, J. S., Finite-Dimentional Variational Inequality and Nonlinear Complementarity Problems: A Survey of Theory, Algorithms, and Applications, Mathematical Programming, Vol. 48, pp. 161-220, 1990. · Zbl 0734.90098
[2]	Pang, J. S., Complementarity Problems, Handbook of Global Optimization, Edited by R. Horst and P. Pardalos, Kluwer Academic Publishers, Norwell, Massachusetts, 1994.
[3]	Ferris, M. C., and Pang, J. S., Engineering and Economic Applications of Complementarity Problems, SIAM Review, Vol. 39, pp. 699-713, 1997. · Zbl 0891.90158
[4]	Fukushima, M., Merit Functions for Variational Inequality and Complementarity Problems, Nonlinear Optimization and Applications, Edited by G. Di Pillo and F. Giannessi, Plenum Press, New York, NY, pp. 155-170, 1996. · Zbl 0996.90082
[5]	Peng, J. M., Equivalence of Variational Inquality Problems to Unconstrained Optimization, Mathematical Programming, Vol. 78, pp. 347-355, 1997. · Zbl 0887.90171
[6]	Yamashita, N., Taji, K., and Fukushima, M., Unconstrained Optimization Reformulations of Variational Inequality Problems, Journal of Optimization Theory and Applications, Vol. 92, pp. 439-456, 1997. · Zbl 0879.90180
[7]	Auslender, A., Optimisation: Méthodes Numériques, Masson, Paris, France, 1976.
[8]	Auchmuty, G., Variational Principles for Variational Inequalities, Numerical Functional Analysis and Optimization, Vol. 10, pp. 863-874, 1989. · Zbl 0678.49010
[9]	Fukushima, M., Equivalent Differentiable Optimization Problems and Descent Methods for Asymmetric Variational Inequality Problems, Mathematical Programming, Vol. 53, pp. 99-110, 1992. · Zbl 0756.90081
[10]	Wu, J. H., Florian, M., and Marcotte, P., A General Descent Framework for the Monotone Variational Inequality Problem, Mathematical Programming, Vol. 61, pp. 281-300, 1993. · Zbl 0813.90111
[11]	Yamashita, N., and Fukushima, M., Equivalent Unconstrained Minimization and Global Error Bounds for Variational Inequality Problems, SIAM Journal on Control and Optimization, Vol. 35, pp. 273-284, 1997. · Zbl 0873.49006
[12]	Peng, J. M., and Fukushima, M., A Hybrid Newton Method for Solving the Variational Inequality Problem via the D-Gap Function, Mathematical Programming, Vol. 86, pp. 367-386, 1999. · Zbl 0939.90023