Abstract
Scalarization of vector problems is an important concept at least from the computational point of view. In this paper, scalarization is applied to Weak Vector Variational Inequalities, and results are established using a symmetric Jacobian condition. New relationships are presented for Vector Variational Inequalities and Vector Optimization problems, and sufficient and necessary conditions for reducing a Weak Vector Variational Inequality to a (scalar) Variational Inequality are discussed. An exact analytical method for solving a special case of the Weak Vector Variational Inequality involving only affine functions via scalarization is proposed.
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References
Chen G.-Y. and Yang X.Q., “The vector complementary problems and its equivalences with weak minimal element”. Jou. of Mathem. Analysis and Appls., Vol. 153, No. 1, 1990, pp. 136–158.
Chen G.-Y. and Yen N.D., “On the Variational inequality model for network equilibrium”. Internal report 3.196 ( 724 ), Department of Mathematics, University of Pisa, 1993.
Clarke F.H., “Optimization and Nonsmooth Analysis”. Wiley, New York (1983).
Gerth (Tammer) Chr. and Weidner P. “Nonconvex separation theorems and some applications in vector optimization”. Jou. of Optimiz. Theory and Appls., Vol. 67 (2), 1990, pp. 297–320.
Giannessi F., “Theorems of alternative, quadratic programs and complementarity problems”. In “Variational Inequalities and Complementarity Problems”, ed. Cottle R.W., Giannessi F., and Lions J.-L., Wiley, New York, 1980, pp. 151–186.
Goh C.J. and Yang X.Q., “Analytic efficient solution set for multi-criteria quadratic programs”. European Jou. of Operations Research, Vol. 92, 1996, pp. 166–181.
Jahn J., “Scalarization in multi-objective optimization”. In “Mathematics of multi-objective optimization”, ed. P. Serafini, Springer-Verlag, New York, 1985, pp. 45–88.
Luc D.T., “Theory of vector optimization”. In “Lecture Notes in Econ. and Mathem. Systems”, No. 319, Springer, Berlin, 1989.
Luenberger D.G., “Linear and Nonlinear Programming”. Second edition, Addison-Wesley, Reading, Massachusetts, 1984.
Mangasarian O.L. “Nonlinear Programming”. McGraw-Hill, New York, 1969.
Nagurney A., “Network Economics: A Variational Inequality Approach”. Kluwer Academic Publishers, 1993.
Ortega J.M. and Rheinboldt W.C., “Iterative solution of nonlinear equations in several variables”. Academic Press, New York, 1970.
Sawaragi Y. Nakayam H. and Tanino T., “Theory of Multiobjective Optimization”. Academic Press, New York, 1985.
Yang X.Q., “Vector variational inequality and its duality”. Nonlinear Analysis, Theory, Methods and Applications, Vol. 21, No. 11, 1993, pp. 869–877.
Yang X.Q. and Goh C.J., “On vector variational inequality. application to vector traffic equilibria”. Jou. of Optimiz. Theory and Appls., Vol. 95, 1997, pp. 431–443.
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© 2000 Kluwer Academic Publishers
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Goh, CJ., Yang, X.Q. (2000). Scalarization Methods for Vector Variational Inequality. 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_12
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DOI: https://doi.org/10.1007/978-1-4613-0299-5_12
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-7985-0
Online ISBN: 978-1-4613-0299-5
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