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Scalarization of vector optimization problems. (English) Zbl 0622.90083
We investigate the scalar representation of vector optimization problems in close connection with monotonic functions. We show that it is possible to construct linear, convex, and quasiconvex representations for linear, convex, and quasiconvex vector problems, respectively. Moreover, for finding all the optimal solutions of a vector problem, it suffices to solve certain scalar representations only. The question of the continuous dependence of the solution set upon the initial vector problems and monotonic functions is also discussed.

MSC:
90C31 Sensitivity, stability, parametric optimization
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[1] Chankong, V., andHaimes, Y. Y.,Multiobjective Decision Making: Theory and Methodology, North-Holland, Amsterdam, Holland, 1983. · Zbl 0622.90002
[2] Warburton, A. R.,Quasiconcave Vector Maximization: Connectedness of the Sets of Pareto-Optimal and Weak Pareto-Optimal Alternatives, Journal of Optimization Theory and Applications, Vol. 40, pp. 537-557, 1983. · Zbl 0496.90073
[3] Pascoletti, A., andSerafini, P.,Scalarizing Vector Optimization Problems, Journal of Optimization Theory and Applications, Vol. 42, pp. 499-524, 1984. · Zbl 0505.90072
[4] Serafini, P.,A Unified Approach for Scalar and Vector Optimization, Proceedings of the Conference on Mathematics of Multiobjective Optimization, CISM, Udine, Italy, 1984. · Zbl 0557.90089
[5] Jahn, J.,Scalarization in Vector Optimization, Mathematical Programming, Vol. 29, pp. 203-218, 1984. · Zbl 0539.90093
[6] Luc, D. T.,Connectedness of Efficient Point Sets in Quasiconcave Vector Maximization, Journal of Mathematical Analysis and Applications (to appear). · Zbl 0615.90087
[7] Krabs, W.,Optimization and Approximation, John Wiley, Chichester, England, 1979. · Zbl 0409.90051
[8] Yu, P. L.,Cone Convexity, Cone Extreme Points, and Nondominated Solutions in Decision Problems with Multiobjectives, Journal of Optimization Theory and Applications, Vol. 14, pp. 319-377, 1974. · Zbl 0268.90057
[9] Naccache, P. H.,Stability in Vector Optimization, Journal of Mathematical Analysis and Applications, Vol. 68, pp. 441-453, 1979. · Zbl 0418.90079
[10] Berge, C.,Topological Spaces, Macmillan, New York, New York, 1963.
[11] Hiriart-Urruty, J. B.,Images of Connected Sets by Semicontinuous Multifunctions, Journal of Mathematical Analysis and Applications, Vol. 111, pp. 407-422, 1985. · Zbl 0578.54013
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