On Some Broad Classes of Vector Optimal Decisions and Their Characterization

  • A. Marzollo
  • W. Ukovich
Part of the International Centre for Mechanical Sciences book series (CISM, volume 211)


We shall consider situations in which some “decision maker” has to choose, in a set of feasible decisions, a decision which may be considered as the “ best” according to some finite set of criteria.


Cooperative Game Optimal Decision Equality Principle Nondominated Solution Marginal Gain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. J.P. Aubin: “A Pareto Minimum Principle”, in Kuhn, Szego (eds.): “ Differential Games and Related Topics”, North-Holland, 1971.Google Scholar
  2. E. Berge, A. Gouila-Houri: “Programming, Games and Transportation Networks”, Wiley, 1965.Google Scholar
  3. A. Blaquière: “ Vector Valued Optimization in Multiplayer Quantitative Games”, this volume, pp. 33–54, 1976.Google Scholar
  4. A. Blaquière, L. Juricek, K. Wiese: “Geometry of Pareto Equilibria and Maximum Principle in N-Person Differential Games” J. of Lath. An. and Appl. vol. 38, 1972.Google Scholar
  5. G. Birkhoff: “Lattice Theory” American Mathematical Society Colloquium Pubbl.ications vol. X XV, 1948.Google Scholar
  6. S.S.L. Chang: “General Theory of Optimal Processes” J. SIAJ. on Control vol. 4, pp. 46–55, 1966.CrossRefzbMATHGoogle Scholar
  7. K.O. Chu: “On the Noninferior Set for the Systems with Vector-Valued Objective Function” IEEE Tr. on Aut. Contr. AC-15, pp. 103124, 1970.Google Scholar
  8. L.O.Da Cunha, E. Polak: “Constrained Minimization Under Vector-Valued Criteria in Finite Dimensional Spaces” J. of Math. An. and Appl. vol. 19, pp. 103–124, 1967.CrossRefzbMATHGoogle Scholar
  9. G. Debreu: “Theory of Value” Wiley, 1959.Google Scholar
  10. D. Gale: “The Theory of Linear Economic Models”, McGraw-Hill, 1960.Google Scholar
  11. A.H. Geoffrion: “Strictly Concave Parametric Programming, Part I: Basic Theory”, Management Science, vol. 13, pp. 244–253, 1966.MathSciNetCrossRefzbMATHGoogle Scholar
  12. A.H. GeofIrion: “Strictly Concave Parametric Programming, Part II: Additional Theory and Computational. Considerations”, Management Science, vol. 13, pp. 359–370, 1967a.MathSciNetCrossRefGoogle Scholar
  13. A.H. Geoffrion: “Solving Bicriterion Mathematical Programs” Op.Res., vol. 15, pp. 39–54, 1967b.MathSciNetCrossRefzbMATHGoogle Scholar
  14. A. Haurie: “On Pareto Optimal Decisions for a Coalition of a Subset of Players” IEEE Tr. on Aut. Contr. AC-18, pp. 144–149, 1973.Google Scholar
  15. A. Haurie, M. C. Delfour: “Individual and Collective Rationality in a Dynamic Pareto Equilibrium”, J.O.T.A. vol. 13, pp. 290–302, 1974.MathSciNetCrossRefzbMATHGoogle Scholar
  16. T. Hoang: “The Farkas-Minkowski Theorem and Extremum Problems”, in J. and M. $os (eds.): “Mathematical Models in Economics”, North-Holland, 1974.Google Scholar
  17. L. Hurwicz: “Programming in Linear Spaces”, in K.J. Arrow, L. Hurwicz, H. Uzawa (eds.): “Studies in Linear and Nonlinear Programming”, Stanford Univ. Press, 1958.Google Scholar
  18. S. Karlin: “Mathematical Methods and Theory in Games, Programming and Economics” vol. 1. Addison-Wesley, 1959.Google Scholar
  19. R.L. Keeney, H. Raiffa: “Decision Analysis with Multiple Conflicting Objectives, Preferences and Value Tradeoffs” II ASA Working Paper WP–75–73, 1975.Google Scholar
  20. A. Klinger: “Vector-Valued Performance Criteria” IEEE Tr. on Aut. Contr. AC-9, pp. 117–118, 1964.Google Scholar
  21. A. Klinger: “Improper Solutions of the Vector Maximum Problem” Op. Res. vol. 15, PP. 570–572, 1967.Google Scholar
  22. T.C. Koopmans (ed.): “Activity Analysis of Production and Allocation”, Cowles Commission Monograph N° 13, Wiley, 1951.Google Scholar
  23. H.W. Kuhn, A.W. Tucker: “Nonlinear Programming” in J. Neyman (ed.): Proc. of the Second Berkeley Symposium on Mathematical Statistics and Probability, Univers. of Calif. Press, pp. 481–491, 1951.Google Scholar
  24. G. Leitmann, S. Rocklin, T.L. Vincent: “A Note on Control-Space Properties of Cooperative Games”, J.O.T.A. vol. 9, 1972.Google Scholar
  25. G. Leitmann, W. Sohmitendorf: “Some Sufficienoy Conditions for Pareto-Optimal Control” J. of Dynamic Systems, Measurement and Control, vol. 95, pp. 356–361, 1973.CrossRefzbMATHGoogle Scholar
  26. G. Leitmann: “Cooperative and Non-Cooperative Many Player Differential Games”, Springer Verlag, 1974.Google Scholar
  27. G. Leitmann: “Cooperative and Non-Cooperative Differential Games’, this volume, pp. 7–32, 1976.Google Scholar
  28. R.D. Luce, H. Raiffa: “Games and Decisions” Wiley, 1957.Google Scholar
  29. K.R. Mao Grimmon: “An Overview of Multiple Objective Decision Making” in J.L. Cochrane, M. Zeleny (eds.): “Multiple Criteria Decision Making”, Univers. of Southern Calif. Press, pp. 18–44, 1973.Google Scholar
  30. O. Mangasarian: “Nonlinear Programming” McGraw-Hill, 1969.Google Scholar
  31. A. Marzollo, W. Ukovich: “A Support Function Approach to the Characterization of the Optimal Gain Vectors in Cooperative Games” Proc. of the 1974 IEEE Conference on Decision and Control, pp. 362–367, 1974.Google Scholar
  32. A. Marzollo, W. Ukovich: “Nondominated Solutions in Cooperative Gamest A Dual Space Approach”, Proc. of the VI IFAC Congress, Boston, 1975.Google Scholar
  33. A. Marzollo, P. Serafini, W. Ukovich: “Decisioni Paretiane e loro determinazione”, in “Teoria dei Sistemi ed Economia” GES -I1 Mulino, 1976.Google Scholar
  34. W. Pareto: “Cours d’Economie Politique” Ronge, 1896.Google Scholar
  35. E. Polak, A.N. Payne: “On Multicriteria Optimization” ERL - M 566 Univ. of Californ., Berkeley, May 1975.Google Scholar
  36. J. Ponstein: “Seven Kinds of Convexity”, SIAM Review, vol. 9, N° 1, 1967.Google Scholar
  37. Z.V. Rekasius, W.E. Schmitendorf: “On the Noninferiority of Nash Equilibrium Solutions” IEEE Tr. on Aut. Contr. AC–16, pp. 170–173, 1971.MathSciNetCrossRefGoogle Scholar
  38. R.T. Rockafellar: “Convex Analysis”, Princeton Press, 1970.Google Scholar
  39. B. Roy: “Problems and Methods with Multiple Objective Functions”, Math. Progr.,vol. 1, pp. 239–266, 1971.CrossRefzbMATHGoogle Scholar
  40. W.E. Schmitendorf: “ Cooperative Games and Vector-Valued Criteria Problems” IEEE Tr. on Aut. Contr. AC–18, pp. 139–144, 1973.MathSciNetCrossRefzbMATHGoogle Scholar
  41. A.K. Sen: “Collective Choice And Social Welfare” Oliver and Boyd, 1970.Google Scholar
  42. S. Smale: “Global Analysis and Economics I: Pareto Optimum and a Generalization of Morse Theory”, in M. Peixoto (ed.): Dynamical Systems, pp. 531–544, Academic Press, 1973.Google Scholar
  43. S. Smale: “Global Analysis and Economics V: Pareto Theory with Constraints” J. of Math. Economics, vol. 1, pp. 213–221, 1974.MathSciNetCrossRefzbMATHGoogle Scholar
  44. W. Stadler: “Preference Optimality and Applications of Pareto Optimality”, this volume.Google Scholar
  45. A.W. Starr, Y.C. Ho: “Nonzero-Sum Differential Games” J.O.T.A., vol. 3, pp. 184–206, 1969a.MathSciNetCrossRefzbMATHGoogle Scholar
  46. A.W. Starr, Y.C. Ho: “Further Properties of Nonzero-Sum Differential Games” J.O.T.A., vol. 3, pp. 207–219, 1969b.MathSciNetCrossRefzbMATHGoogle Scholar
  47. T.L. Vincent, G. Leitmann: “Control-Space Properties of Cooperative Games” J.O.T.A., vol. 6, pp. 91–104, 1970.MathSciNetCrossRefzbMATHGoogle Scholar
  48. Y.H. Wan: “On Local Pareto Optimum” J. of Math. Economics, vol. 2, 1975.Google Scholar
  49. P.L. Yu: “Cone Convexity, Cone Extreme Points and Nondominated Solutions in Decision Problems with Multiobjectives”, University of Rochester, Center for System Science, Report 72–02, 1972, reprinted in J.O.T.A., 1974.Google Scholar
  50. P.L. Yu: “Domination Structures and Nondominated Solutions”, this volume.Google Scholar
  51. L.A. Zadeh: “Optimality and Non-Scalar-Valued Performance Criteria” IEEE Tr. on Aut. Contr. AC–8, pp. 59–60, 1963.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 1975

Authors and Affiliations

  • A. Marzollo
    • 1
    • 2
  • W. Ukovich
    • 1
    • 2
  1. 1.Electrical Engineering DepartmentUniversity of TriesteItaly
  2. 2.International Centre for Mechanical SciencesUdineItaly

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