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On Pareto optima, the Fermat-Weber problem, and polyhedral gauges. (English) Zbl 0706.90066
This paper deals with problems of scalarization in vector optimization, in the framework of location theory with respect to objective functions involving polyhedral gauges. A nice concept (namely, the so-called diff- max property) is developed and used to prove that each Pareto optimum is properly efficient (in other words, each Pareto optimum is the solution to a Fermat-Weber problem, with strictly positive coefficients). This concept is also used to characterize polyhedral gauges. Finally, practical rules are given in order to obtain the whole set of efficient points of a location problem involving polyhedral gauges. However, an efficient implementation of the described procedure does not seem very easy.
Reviewer: P.Loridan

90C29Multi-objective programming; goal programming
90B85Continuous location
Full Text: DOI
[1] C. Berge,Espaces Topologiques, Fonctions Multivoques (Dunod, Paris, 1959). · Zbl 0088.14703
[2] J.M. Borwein, ”Proper efficient points for maximization with respect to cones,”SIAM Journal on Control and Optimization 15 (1977) 57--63. · Zbl 0369.90096 · doi:10.1137/0315004
[3] G.R. Britan and T.L. Magnanti, ”The structure of admissible points with respect to cone dominance,”Journal of Optimization Theory and Applications 29 (1979) 573--614. · Zbl 0389.52021 · doi:10.1007/BF00934453
[4] A. Brønsted,An Introduction to Convex Polytopes (Springer, New York, 1983). · Zbl 0509.52001
[5] L.G. Chalmet, R.L. Francis and A. Kolen, ”Finding efficient solutions for rectilinear distance location problem efficiently,”European Journal of Operational Research 6 (1981) 117--124. · Zbl 0451.90037 · doi:10.1016/0377-2217(81)90197-1
[6] R. Durier, ”Sets of efficiency in a normed space and inner product,” in: J. Jahn and W. Krabs, eds.,Recent advances and historical development of vector optimization, Lecture Notes in Economics and Mathematical Systems, Vol. 294 (Springer, Berlin, 1987) pp. 114--128.
[7] R. Durier, ”Meilleure approximation en norme vectorielle et théorie de la localisation,”RAIRO Modelisation Mathématique et Analyse Numérique 21 (1987) 605--626. · Zbl 0649.41019
[8] R. Durier, ”Weighting factor results in vector optimization,”Journal of Optimization Theory and Applications 58 (1988) 411--430. · Zbl 0628.90075 · doi:10.1007/BF00939390
[9] R. Durier and C. Michelot, ”Geometrical properties of the Fermat-Weber problem,”European Journal of Operational Research 20 (1985) 332--343. · Zbl 0564.90013 · doi:10.1016/0377-2217(85)90006-2
[10] R. Durier and C. Michelot, ”Sets of efficient points in a normed space,”Journal of Mathematical Analysis and Applications 117 (1986) 506--528. · Zbl 0605.49020 · doi:10.1016/0022-247X(86)90237-4
[11] D. Gale,The Theory of Linear Economic Models (McGraw-Hill, New York, 1960). · Zbl 0114.12203
[12] A.M. Geoffrion, ”Proper efficiency and the theory of vector maximization,”Journal of Mathematical Analysis and Applications 22 (1968) 618--630. · Zbl 0181.22806 · doi:10.1016/0022-247X(68)90201-1
[13] P. Hansen, J. Perreur and J.F. Thisse, ”Location theory, dominance and convexity: some further results,”Operations Research 28 (1980) 1241--1250. · Zbl 0449.90027 · doi:10.1287/opre.28.5.1241
[14] P. Hansen and J.F. Thisse, ”Recent advances in continuous location theory,”Sistemi Urbani 1 (1983) 33--54.
[15] J.-B. Hiriart-Urruty, ”Images of connected sets by semicontinuous multifunctions,”Journal of Mathematical Analysis and Applications 111 (1985) 407--422. · Zbl 0578.54013 · doi:10.1016/0022-247X(85)90225-2
[16] S. Karlin,Mathematical Methods and Theory in Games, Programming, and Economics, Vol I (Addison-Wesley, Reading, MA, 1959). · Zbl 0139.12704
[17] H.W. Kuhn, ”A pair of dual nonlinear programs,” in: J. Abadie, ed.,Methods of Nonlinear Programming (North-Holland, Amsterdam, 1967) pp 37--54. · Zbl 0183.22804
[18] H.W. Kuhn, ”A note on Fermat’s problem,”Mathematical Programming 4 (1973) 98--107. · Zbl 0255.90063 · doi:10.1007/BF01584648
[19] L. McLinden, ”Polyhedral extension of some theorems of linear programming,”Mathematical Programming 24 (1982) 162--176. · Zbl 0495.90056 · doi:10.1007/BF01585102
[20] H. Minkowski,Theorie der Konvexen Körper, Gesammelte Abhandlugen, Vol. II (Teubner, Berlin, 1911).
[21] P.H. Naccache, ”Connectedness of the set of nondominated outcomes in multicriteria optimization,”Journal of Optimization Theory and Applications 25 (1978) 459--467. · Zbl 0363.90108 · doi:10.1007/BF00932907
[22] F. Plastria, ”Continuous location problems and cutting plane algorithms,” Thesis, Vrije Universiteit Brussel (Brussels, 1983).
[23] R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970). · Zbl 0193.18401
[24] J.F. Thisse, J.E. Ward and R.E. Wendell, ”Some properties of location problems with block and round norms,”Operations Research 32 (1984) 1309--1327. · Zbl 0557.90023 · doi:10.1287/opre.32.6.1309
[25] J.E. Ward and R.E. Wendell, ”Characterizing efficient points in location problems under the one-infinity norm,” in: J.F. Thisse and H.G. Zoller, eds.,Locational analysis of public facilities, Studies in Mathematical and Managed Economics, Vol. 31 (North-Holland, Amsterdam, 1983) pp. 413--429.
[26] A. Weber,Über den Standort der Industrien (Tübingen, 1909). [English translation:The Theory of the Location of Industries (Chicago University Press, Chicago, IL, 1929).]
[27] R.E. Wendell and A.P. Hurter, ”Location theory, dominance and convexity,”Operations Research 21 (1973) 314--321. · Zbl 0265.90040 · doi:10.1287/opre.21.1.314
[28] R.E. Wendell, A.P. Hurter and T.J. Lowe, ”Efficient points in location problems,”AIEE Transactions 9 (1973) 238--246.
[29] P.L. Yu, ”Cone convexity, cone extreme points and nondominated solutions in decision problems with multiobjectives,”Journal of Optimization Theory and Applications 14 (1974) 319--377. · Zbl 0268.90057 · doi:10.1007/BF00932614
[30] P.L. Yu and M. Zeleny, ”The set of all nondominated solutions in linear cases and a multicriteria simplex method,”Journal of Mathematical Analysis and Applications 49 (1975) 430--468. · Zbl 0313.65047 · doi:10.1016/0022-247X(75)90189-4