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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.

MSC:

90C31 Sensitivity, stability, parametric optimization

Cite

References:

[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 ·doi:10.1007/BF00932518
[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 ·doi:10.1007/BF00933455
[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 ·doi:10.1137/0315004
[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 ·doi:10.1007/BF00933599
[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 ·doi:10.1016/0022-247X(68)90201-1
[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 ·doi:10.1007/BF00932935
[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 ·doi:10.1007/BF02547750
[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 ·doi:10.1007/BF00933793
[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 ·doi:10.1007/BF02243436
[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 ·doi:10.1287/mnsc.19.8.936
[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 ·doi:10.1007/BF00934871
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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