Scalarization in vector optimization. (English) Zbl 0539.90093

A scalarization of a general nonconvex vector optimization problem is investigated. Scalarization means the replacement of the vector optmization problem by a scalar optimization problem. Under suitable assumptions it is shown that an optimal solution of the considered vector optimization problem is also a solution of an appropriate approximation problem. With the aid of this theory a complete characterization of minimal and weakly minimal elements of a nonempty nonconvex subset of a partially ordered real linear space is presented. Moreover, these results are applied to general vector approximation problems.


90C31 Sensitivity, stability, parametric optimization
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[1] A. Bacopoulos, G. Godini and I. Singer, ”On best approximation in vector-valued norms”,Colloquia Mathematica Societatis János Bolyai 19 (1978) 89–100. · Zbl 0425.41026
[2] 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. · Zbl 0393.90089
[3] 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. · Zbl 0482.41026
[4] A. Bacopoulos and I. Singer, ”On convex vectorial optimization in linear spaces”,Journal of Optimization Theory and Applications 21 (1977) 175–188. · Zbl 0326.90052
[5] A. Bacopoulos and I. Singer, ”Errata corrige: On vectorial optimization in linear spaces”,Journal of Optimization Theory and Applications 23 (1977) 473–476. · Zbl 0326.90052
[6] J. Borwein, ”Proper efficient points for maximizations with respect to cones”,SIAM Journal on Control and Optimization 15 (1977) 57–63. · Zbl 0369.90096
[7] J.M. Borwein, ”The geometry of Pareto efficiency over cones”,Mathematische Operationsforschung und Statistik 11 (1980) 235–248. · Zbl 0447.90077
[8] 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. · Zbl 0269.90040
[9] W. Dinkelbach and W. Dürr, ”Effizienzaussagen bei Ersatzprogrammen zum Vektormaximumproblem”Operations Research Verfahren XII (1972) 69–77. · Zbl 0253.90056
[10] N. Dunford and J.T. Schwartz,Linear operators, Part I (Interscience Publishers, 1957). · Zbl 0128.34803
[11] W.B. Gearhart, ”Compromise solutions and estimation of the noninferior set”,Journal of Optimization Theory and Applications 28 (1979) 29–47. · Zbl 0422.90074
[12] A.M. Geoffrion, ”Proper efficiency and the theory of vector maximazation”,Journal of Mathematical Analysis and Applications 22 (1968) 618–630. · Zbl 0181.22806
[13] S.C. Huang, ”Note on the mean-square strategy of vector values objective function”,Journal of Optimization Theory and Applications 9 (1972) 364–366. · Zbl 0222.49004
[14] 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.
[15] L. Kantorovitch, ”The method of successive approximations for functional equations”,Acta Mathematica 71 (1939) 63–97. · Zbl 0021.13604
[16] W. Krabs,Optimization and approximation (John Wiley & Sons, New York, 1979). · Zbl 0409.90051
[17] J.G. Lin, ”Maximal vectors and multi-objective optimization”,Journal of Optimization Theory and Applications 18 (1976) 41–64. · Zbl 0298.90056
[18] R. Reemtsen, ”On level sets and an approximation problem for the numerical solution of a free boundary problem”,Computing 27 (1981) 27–35. · Zbl 0457.65042
[19] S. Rolewicz, ”On a norm scalarization in infinite dimensional Banach spaces”,Control and Cybernetics 4 (1975) 85–89. · Zbl 0334.49015
[20] M.E. Salukvadze, ”Optimization of vector functionals” (in Russian).Automatika i Telemekhanika 8 (1971) 5–15. · Zbl 0236.49019
[21] W. Vogel,Vektoroptimierung in Produkträumen (Verlag Anton Hain, Mathematical Systems in Economics 35, 1977). · Zbl 0392.90044
[22] 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. · Zbl 0422.90075
[23] 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).
[24] A.P. Wierzbicki, ”Basic properties of scalarizing functionals for multiobjective optimization”,Mathematische Operationsforschung und Statistik 8 (1977) 55–60.
[25] 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.
[26] 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.
[27] P.L. Yu, ”A class of solutions for group decision problems”,Management Science 19 (1973) 936–946. · Zbl 0264.90008
[28] P.L. Yu and G. Leitmann, ”Compromise solutions, domination structures, and Salukvadze’s solution”,Journal of Optimization Theory and Applications 13 (1974) 362–378. · Zbl 0362.90111
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