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On Gap Functions for Quasi‐Variational Inequalities

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  • Kouichi Taji

Abstract

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.

Suggested Citation

  • Kouichi Taji, 2008. "On Gap Functions for Quasi‐Variational Inequalities," Abstract and Applied Analysis, John Wiley & Sons, vol. 2008(1).
    • Handle: RePEc:wly:jnlaaa:v:2008:y:2008:i:1:n:531361
    DOI: 10.1155/2008/531361
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    References listed on IDEAS

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    1. N. Yamashita & K. Taji & M. Fukushima, 1997. "Unconstrained Optimization Reformulations of Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 92(3), pages 439-456, March.
    2. Harker, Patrick T., 1991. "Generalized Nash games and quasi-variational inequalities," European Journal of Operational Research, Elsevier, vol. 54(1), pages 81-94, September.
    3. William Hogan, 1973. "Directional Derivatives for Extremal-Value Functions with Applications to the Completely Convex Case," Operations Research, INFORMS, vol. 21(1), pages 188-209, February.
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