For variational inequalities, various merit functions, such as the gap function, the regularized gap function, the D-gap function and so on, have been proposed. These functions lead to equivalent optimization formulations and are used to optimization-based methods for solving variational inequalities. In this paper, we extend the regularized gap function and the D-gap functions for a quasi-variational inequality, which is a generalization of the variational inequality and is used to formulate generalized equilibrium problems. These extensions are shown to formulate equivalent optimization problems for quasi-variational inequalities and are shown to be continuous and directionally differentiable.
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription.
Read more about accessing full-text
References
P. T. Harker, ``Generalized Nash games and quasi-variational inequalities,'' European Journal of Operational Research, vol. 54, no. 1, pp. 81--94, 1991.
J.-S. Pang and M. Fukushima, ``Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games,'' Computational Management Science, vol. 2, no. 1, pp. 21--56, 2005.
M. Fukushima, ``Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems,'' Mathematical Programming, vol. 53, no. 1, pp. 99--110, 1992.
J.-M. Peng, ``Equivalence of variational inequality problems to unconstrained minimization,'' Mathematical Programming, vol. 78, no. 3, pp. 347--355, 1997.
N. Yamashita, K. Taji, and M. Fukushima, ``Unconstrained optimization reformulations of variational inequality problems,'' Journal of Optimization Theory and Applications, vol. 92, no. 3, pp. 439--456, 1997.
K. Taji, M. Fukushima, and T. Ibaraki, ``A globally convergent Newton method for solving strong monotone variational inequalities,'' Mathematical Programming, vol. 58, no. 3, pp. 369--383, 1993.
J.-M. Peng and M. Fukushima, ``A hybrid Newton method for solving the variational inequality problem via the D-gap function,'' Mathematical Programming, vol. 86, no. 2, pp. 367--386, 1999.
M. Fukushima, ``A class of gap functions for quasi-variational inequality problems,'' Journal of Industrial and Management Optimization, vol. 3, no. 2, pp. 165--171, 2007.
J. H. Wu, M. Florian, and P. Marcotte, ``A general descent framework for the monotone variational inequality problem,'' Mathematical Programming, vol. 61, no. 3, pp. 281--300, 1993.
W. Hogan, ``Point-to-set maps in mathematical programming,'' SIAM Review, vol. 15, pp. 591--603, 1973.
K. Taji and M. Fukushima, ``A new merit function and a successive quadratic programming algorithm for variational inequality problems,'' SIAM Journal on Optimization, vol. 6, no. 3, pp. 704--713, 1996.
W. Hogan, ``Directional derivatives for extremal-value functions with applications to the completely convex case,'' Operations Research, vol. 21, pp. 188--209, 1973.
