Abstract
As it was shown by C. Malivert (1996) and other Authors, in a Linear Fractional Vector Optimization Problem (for short, LFVOP) any point satisfying the first-order necessary optimality condition (a stationary point) is a solution. Therefore, solving such a problem is equivalent to solve a monotone affine vector variational inequality of a special type. This observation allows us to apply the existing results on monotone affine variational inequality to establish some facts about connectedness and stability of the solution sets in LFVOP. In particular, we are able to solve a question raised by E. U. Choo and D. R. Atkins (1983) by proving that the set of all the efficient points (Pareto solutions) of a LFVOP with a bounded constraint set is connected.
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References
Bednarczuk E.M., “Berge-Type Theorems for Vector Optimiz Problems”. Optimiz., Vol. 32, 1995, pp. 373–384.
Choo E.U. and Atkins D.R., “Bicriteria Linear Fractional Programming”. Jou. of Optimiz. Theory and Appls., Vol. 36, 1982, pp. 203–220.
Choo E.U. and Atkins D.R., “Connectedness in Multiple Linear Fractional Programming”. Manag. Science, Vol. 29, 1983, pp. 250–255.
Craven B.D., “Fractional Programming”. Heldermann-Verlag, Berlin, 1988.
Giannessi F., “Theorems of Alternative, Quadratic Programs and Complementarity Problems”. In “Variational Inequality and Complementarity Problems” (Edited by R. W. Cottle, F. Giannessi and J.-L. Lions), Wiley, New York, 1980, pp. 151–186.
Hu Y.D. and Sun E.J., “Connectedness of the Efficient Set in Strictly Quasiconcave Vector Maximization”. Jou. of Optimiz. Theory and Appls., Vol. 78, 1993, pp. 613–622.
Kinderlehrer D. and Stampacchia G., “An Introduction to Variational Inequalities and Their Appls.”. Academic Press, New York, 1980.
Malivert C., “Multicriteria Fractional Programming”. ( Manuscript, September 1996 )
Lee G.M., Kim D.S., Lee B.S. and Yen N.D., “Vector Variational Inequalities as a Tool for Studying Vector Optimization Problems”. Nonlinear Analysis, Vol. 34, 1998, pp. 745–765.
Luc D.T., “Theory of Vector Optimiz.”. Springer-Verlag, Berlin, 1989.
Penot J.-P. and Sterna-Karwat A., “Parameterized Multicriteria Optimiz.: Continuity and Closedness of Optimal Multifunctions”. Jou. of Mathem. Analysis and Appls., Vol. 120, 1986, pp. 150–168.
Robinson S.M., “Generalized Equations and Their Solutions, Part I: Basic Theory”. Mathem. Programming Study, Vol. 10, 1979, pp. 128–141.
Steuer R.E., “Multiple Criteria Optimiz.: Theory, Computation and Application”. J. Wiley & Sons, New York, 1986.
Wantao F. and Kunping Z., “Connectedness of the Efficient Solution Sets for a Strictly Path Quasiconvex Programming Problem”. Nonlinear Analysis: Theory, Methods and Appls., Vol. 21, 1993, pp. 903–910.
Warburton A.R., “Quasiconcave Vector Maximization: Connectedness of the Sets of Pareto-Optimal and Weak Pareto-Optimal Alternatives”. Jou. of Optimiz. Theory and Appls., Vol. 40, 1983, pp. 537–557.
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© 2000 Kluwer Academic Publishers
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Yen, N.D., Phuong, T.D. (2000). Connectedness and Stability of the Solution Sets in Linear Fractional Vector Optimization Problems. 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_29
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DOI: https://doi.org/10.1007/978-1-4613-0299-5_29
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