Abstract
We extend the theory of gap functions for scalar variational inequality problems (see [1,8]) to the case of vector variational inequality. The gap functions for vector variational inequality are defined as set-valued mappings. The significance of the gap function is interpreted in terms of the inverse vector variational inequality. Convexity properties of these set-valued mappings are studied under different assumptions.
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References
Auslander A., “Optimisation: Méthodes Numériques”. Masson, Paris, 1976.
Borwein J.M., “Generic Differentiability of Order-bounded Convex Operators”. Jou. of the Australian Mathem. Soc., Series B, Vol. 28, 1986, pp. 22–29.
Chen G.-Y. and Cheng Q.M, “Vector variational inequality and vector Optimization”. Lecture Notes in Econ. and Mathem. Systems, Vol. 285, 1988, pp. 408–416.
Chen G.-Y. and Craven B.D., “A Vector Variational Inequality and Optimization over an Efficient Set”. ZOR-Mathematical Models of Operations Research, Vol. 341, 1990, pp. 1–12.
Chen G-Y. and Yang X.Q., “The vector complementarity problem and its equivalences with weak minimal element”. Jou. of Mathem. Analysis and Appls., Vol. 153, 1990, pp. 136–158.
Chen G.-Y. and Yen D., “Equilibrium Problems on a Network with Vector-Valued Cost Functions”. Tech. Paper n. 3.191 ( 717 ) Dept. of Mathem., Univ. of Pisa, Italy, 1992.
Giannessi F., “Theorems of alternative, quadratic programs and complementarity problems’. In ”Variational Inequality Complementary Problems“ (R.W. Cottle, F. Giannessi, and J.-L. Lions, Eds.), Wiley, New York, 1980, pp. 151–186.
Giannessi F., “On some connections among variational inequalities, combinatorial and continuous optimization”. Annals of Operations Research, Vol. 58, 1995, pp. 181–200.
Giannessi F., “On Minty Variational Principle”. In “New Trends in Mathematical Programming”, Kluwer, 1997, pp. 93–99.
Hearn D.W., “The gap function of a convex program”, Operations Research Letter, Vol. 1, 1982, pp. 67–71.
Isermann H., “Duality in Multi-Objective Linear Programming”. In “Multiple Criteria Problem Solving”, S. Zions (ed.), Springer-Verlag, 1978.
Jahn J., “Scalarization in Multi-Objective Optimization”. In “Mathematics of Multi-Objective Optimization”, ed. P. Serafini, Springer-Verlag, New York, 1984, pp. 45–88.
Mosco U., “Dual Variational Inequalities”. Jou. of Mathem. Analysis and Appls., Vol. 40, 1972, pp. 202–206.
Sawaragi, Y., Nakayama and Tanino, T., “Theory of Multiobjective Optimziation”. Academic Press, New York, 1985.
Yang X.Q., “A Hahn Banach Theorem in Ordered Linear Spaces and Its Applications”. Optimization, Vol. 23, 1992, pp. 1–9.
Yang X.Q., “Vector Variational Inequality and its Duality”. Nonlinear Analysis. Theory, Methods and Appls., Vol. 21, 1993, pp. 869–877.
Yang X.Q., “Vector Variational Inequality and Vector Pseudolinear Optimization”. Jou. of Optimiz. Theory and Appls., Vol. 95, 1997, pp. 729–734.
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© 2000 Kluwer Academic Publishers
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Chen, Gy., Goh, CJ., Yang, X.Q. (2000). On Gap Functions for Vector Variational Inequalities. In: Giannessi, F. (eds) Vector Variational Inequalities and Vector Equilibria. Nonconvex Optimization and Its Applications, vol 38. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0299-5_4
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DOI: https://doi.org/10.1007/978-1-4613-0299-5_4
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