Domination Structures and Nondominated Solutions

  • Po-Lung Yu
Part of the Mathematical Concepts and Methods in Science and Engineering book series (MCSENG, volume 30)


In Chapter 3 we discussed that “Pareto preference” is based on the assumption that “more is better,” with no other preference information assumed. This is the simplest kind of preference. Recall that Pareto preference is not a weak order because the induced indifference relation, {~}, is not transitive. On the other hand, from Chapters 4–6, we know that a value function or distance function associated with a compromise solution must assume that the corresponding preference {≻} is a weak order and the induced indifference curves must contain a countable dense subset with respect to {≻}. The assumptions for {≻} to have a value function representation certainly are very restrictive, as discussed in Chapter 5. The gap between the assumption of Pareto preference and that of the preference having a value function representation is very large.


Convex Cone Maximum Point Prove Theorem Tangent Cone Constraint Qualification 
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Suggested Reading

  1. 40.
    Benson, H. P., An improved definition of proper efficiency for vector maximization with respect to cones, J. Math Anal. AppL71, 232–241 (1979).MathSciNetMATHCrossRefGoogle Scholar
  2. 51.
    Bitran, G. R., Duality for nonlinear multiple-criteria optimization problems, J. Optim. Theory Appl.35, 367–401 (1981).MathSciNetMATHCrossRefGoogle Scholar
  3. 58.
    Borwein, J., Proper efficient points for maximizations with respect to cones, SIAM J. Control Optim.15, 57–63 (1977).MATHCrossRefGoogle Scholar
  4. 195.
    Hartley, R., On cone-efficiency, cone-convexity and cone-compactness, SIAM J. AppL Math. 34, 211–222 (1978).MathSciNetMATHGoogle Scholar
  5. 198.
    Hazen, G. B., and Morin, T. L., Steepest ascent algorithms for nonconical multiple objective programming, Northwestern University, Department of Industrial Engineering and Management Science, Working Paper, Evanston, Illinois, 1982.Google Scholar
  6. 199.
    Hazen, G. B., and Morin, T. L., Optimality conditions in nonconical multiple-objective programming, J. Optim. Theory Appl.40, 25–59 (1983).MathSciNetMATHCrossRefGoogle Scholar
  7. 205.
    Henig, M. I., Proper efficiency with respect to cones, J. Optim. Theory AppL36, 387–407 (1982).MathSciNetMATHCrossRefGoogle Scholar
  8. 233.
    Isermann, H., Duality in multiple objective linear programming, Multiple Criteria Problem Solving (S. Zionts, ed.), Springer-Verlag, New York (1978), pp. 274–285.CrossRefGoogle Scholar
  9. 235.
    Jahn, J., Duality theory for vector optimization problems in normed linear spaces, Reprint No. 534, Fachbereich Mathematik, Technische Hochschule, Darmstadt, West Germany, 1980.Google Scholar
  10. 265.
    Kornbluth, J. S. H., Duality, indifference and sensitivity analysis in multiple objective linear programming, Oper. Res. Q.25, 599–614 (1974).MathSciNetMATHCrossRefGoogle Scholar
  11. 287.
    Leitmann, G., An Introduction to Optimal Control, McGraw-Hill, New York (1966).MATHGoogle Scholar
  12. 301.
    Luce, R. D., and Raiffa, H., Games and Decisions, Wiley, New York (1967).Google Scholar
  13. 317.
    Nakayama, H., A geometric consideration on duality in vector optimization, J. Optim. Theory Appl.44, 625–655 (1984).MathSciNetMATHCrossRefGoogle Scholar
  14. 324.
    Nieuwenhuis, J. W., Some results about nondominated solutions, J. Optim. Theory Appl.36, 289–301 (1982).MathSciNetMATHCrossRefGoogle Scholar
  15. 358.
    Ponstein, J., On the dualization of multiobjective optimization problems, University of Groningen, Econometric Institute, Working Paper, Groningen, The Netherlands, 1982.Google Scholar
  16. 438.
    Stoer, J., and Witzgall, C., Convexity and Optimization in Finite Dimensions I, Springer-Verlag, New York (1970).Google Scholar
  17. 443.
    Tamura, K., A method for constructing the polar cone of a polyhedral cone, with applications to linear multicriteria decision problems, J. Optim. Theory Appl.19, 547–564 (1976).MATHCrossRefGoogle Scholar
  18. 445.
    Tanino, T., and Sawaragi, Y., Duality theory in multiobjective programming, J. Optim. Theory Appl.27, 509–529 (1979).MathSciNetMATHCrossRefGoogle Scholar
  19. 446.
    Tanino, T., and Sawaragi, Y., Stability of nondominated solutions in multicriteria decision making, J. Optim. Theory Appl.30, 229–253 (1980).MathSciNetMATHCrossRefGoogle Scholar
  20. 481.
    White, D. J., Optimality and Efficiency, Wiley, New York (1982).MATHGoogle Scholar
  21. 492.
    Yu, P. L., Nondominated investment policies in stock market, Systems Analysis Program, The University of Rochester, The Graduate School of Management, Rochester, New York, 1972.Google Scholar
  22. 494.
    Yu, P. L., Introduction to domination structures in multicriteria decision problems, in Multicriteria Decision Making, J. L. Cochrane and M. Zeleny (eds.), University of South Carolina Press, Columbia (1973) pp. 249–261.Google Scholar
  23. 495.
    Yu, P. L., Cone convexity, cone extreme points and nondominated solutions in decision problems with multiobjectives, J. Optim. Theory Appl.14, 319–377 (1974).MATHCrossRefGoogle Scholar
  24. 502.
    Yu, P. L., Dissolution of fuzziness for better decisions-Perspective and techniques, TIMS Studies in Management Sciences, Vol. 20, M. J. Zimmerman, L. A. Zadeh and B. R. Gains (eds.), pp. 171–207, North-Holland, New York (1984).Google Scholar
  25. 508.
    Yu, P. L., and Zeleny, M., The set of all nondominated solutions in the linear case and a multicriteria simplex method, J. Math. Anal. App!. 49, 430–468 (1974).MathSciNetCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1985

Authors and Affiliations

  • Po-Lung Yu
    • 1
  1. 1.University of KansasLawrenceUSA

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