Abstract
In this paper some scalar optimization problems are presented whose optimal solutions are also solutions of a general vector optimization problem. This will be done for weakly minimal and minimal solutions, respectively. Finally the results will be applied to a certain class of approximation problems.
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References
A. Bacopoulos, G. Godini and I. Singer, “On best approximation in vector-valued norms”,Colloquia Mathematica Societatis János Bolyai 19 (1978) 89–100.
A. Bacopoulos, G. Godini and I. Singer, “Infima of sets in the plane and applications to vectorial optimization”,Revue Roumaine de Mathématiques Pures et Appliquées 23 (1978) 343–360.
A. Bacopoulos, G. Godini and I. Singer, “On infima of sets in the plane and best approximation, simultaneous and vectorial, in a linear space with two norms”, in: J. Frehse, D. Pallaschke and U. Trottenberg, eds.,Special topics of applied mathematics (North-Holland, Amsterdam, 1980), pp. 219–239.
A. Bacopoulos and I. Singer, “On convex vectorial optimization in linear spaces”,Journal of Optimization Theory and Applications 21 (1977) 175–188.
A. Bacopoulos and I. Singer, “Errata corrige: On vectorial optimization in linear spaces”,Journal of Optimization Theory and Applications 23 (1977) 473–476.
J. Borwein, “Proper efficient points for maximizations with respect to cones”,SIAM Journal on Control and Optimization 15 (1977) 57–63.
J.M. Borwein, “The geometry of Pareto efficiency over cones”,Mathematische Operationsforschung und Statistik 11 (1980) 235–248.
W. Dinkelbach, “Über eine Lösungsansatz zum Vektormaximumproblem”, in: M. Beckmann, ed.,Unternehmensforschung Heute (Springer, Lecture Notes in Operations Research and Mathematical Systems No. 50, 1971), pp. 1–13.
W. Dinkelbach and W. Dürr, “Effizienzaussagen bei Ersatzprogrammen zum Vektormaximumproblem”Operations Research Verfahren XII (1972) 69–77.
N. Dunford and J.T. Schwartz,Linear operators, Part I (Interscience Publishers, 1957).
W.B. Gearhart, “Compromise solutions and estimation of the noninferior set”,Journal of Optimization Theory and Applications 28 (1979) 29–47.
A.M. Geoffrion, “Proper efficiency and the theory of vector maximazation”,Journal of Mathematical Analysis and Applications 22 (1968) 618–630.
S.C. Huang, “Note on the mean-square strategy of vector values objective function”,Journal of Optimization Theory and Applications 9 (1972) 364–366.
L. Hurwicz, “Programming in linear spaces”, in: K.J. Arrow, L. Hurwicz and H. Uzawa, eds.,Studies in linear and non-linear programming (Stanford University Press, Stanford, 1958). 38–102.
L. Kantorovitch, “The method of successive approximations for functional equations”,Acta Mathematica 71 (1939) 63–97.
W. Krabs,Optimization and approximation (John Wiley & Sons, New York, 1979).
J.G. Lin, “Maximal vectors and multi-objective optimization”,Journal of Optimization Theory and Applications 18 (1976) 41–64.
R. Reemtsen, “On level sets and an approximation problem for the numerical solution of a free boundary problem”,Computing 27 (1981) 27–35.
S. Rolewicz, “On a norm scalarization in infinite dimensional Banach spaces”,Control and Cybernetics 4 (1975) 85–89.
M.E. Salukvadze, “Optimization of vector functionals” (in Russian).Automatika i Telemekhanika 8 (1971) 5–15.
W. Vogel,Vektoroptimierung in Produkträumen (Verlag Anton Hain, Mathematical Systems in Economics 35, 1977).
W. Vogel, “Halbnormen und Vektoroptimierung”, in: H. Albach, E. Helmstädter and R. Henn, eds.,Quantitative Wirtschaftsforschung-Wilhelm Krelle zum 60. Geburtstag (Tübingen 1977), 703–714.
A.P. Wierzbicki, Penalty methods in solving optimization problems with vector performance criteria (Technical Report of the Institute of Automatic Control, TU of Warsaw, 1974).
A.P. Wierzbicki, “Basic properties of scalarizing functionals for multiobjective optimization”,Mathematische Operationsforschung und Statistik 8 (1977) 55–60.
A.P. Wierzbicki, “The use of reference objectives in multi-objective optimization”, in: G. Fandel and T. Gal, eds.,Multiple criteria decision making-Theory and application (Springer, Lecture Notes in Economics and Mathematical Systems No. 177, 1980), pp. 468–486.
A.P. Wierzbicki, “A mathematical basis for satisficing decision making”, in: J.N. Morse,Organizations: Multiple agents with multiple criteria (Springer, Lecture Notes in Economics and Mathematical Systems No. 190, 1981), pp. 465–485.
P.L. Yu, “A class of solutions for group decision problems”,Management Science 19 (1973) 936–946.
P.L. Yu and G. Leitmann, “Compromise solutions, domination structures, and Salukvadze's solution”,Journal of Optimization Theory and Applications 13 (1974) 362–378.
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Johannes, J. Scalarization in vector optimization. Mathematical Programming 29, 203–218 (1984). https://doi.org/10.1007/BF02592221
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DOI: https://doi.org/10.1007/BF02592221