Skip to main content
Springer Nature Link
Log in

The structure of admissible points with respect to cone dominance

  • Contributed Papers
  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We study the set of admissible (Pareto-optimal) points of a closed, convex setX when preferences are described by a convex, but not necessarily closed, cone. Assuming that the preference cone is strictly supported and making mild assumptions about the recession directions ofX, we extend a representation theorem of Arrow, Barankin, and Blackwell by showing that all admissible points are either limit points of certainstrictly admissible alternatives or translations of such limit points by rays in the closure of the preference cone. We also show that the set of strictly admissible points is connected, as is the full set of admissible points.

Relaxing the convexity assumption imposed uponX, we also consider local properties of admissible points in terms of Kuhn-Tucker type characterizations. We specify necessary and sufficient conditions for an element ofX to be a Kuhn-Tucker point, conditions which, in addition, provide local characterizations of strictly admissible points.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Debreu, G.,Representation of a Preference Ordering by a Numerical Function, Decision Processes, Edited by R. M. Thrall, C. H. Coombs, and R. L. Davis, John Wiley and Sons, New York, New York, 1954.

    Google Scholar 

  2. Debreu, G.,Theory of Value, John Wiley and Sons, New York, New York, 1959.

    Google Scholar 

  3. Bowen, R.,A New Proof of a Theorem in Utility Theory, International Economic Review, Vol. 9, p. 374, 1968.

    Google Scholar 

  4. Arrow, K. J., andHahn, F. H.,General Competitive Equilibrium, Holden-Day, San Francisco, California, 1971.

    Google Scholar 

  5. Keeney, R., andRaiffa, H.,Decisions with Multiple Objectives: Preferences and Value Trade-offs, John Wiley and Sons, New York, New York, 1976.

    Google Scholar 

  6. MacCrimmon, K. R.,An Overview of Multiple-Objective Decision-Making, Multiple Criteria Decision Making, Edited by J. L. Cochrane and M. Zeleny, University of South Carolina Press, Columbus, South Carolina, 1973.

    Google Scholar 

  7. Wald, A.,Statistical Design Functions, John Wiley and Sons, New York, New York, 1950.

    Google Scholar 

  8. Arrow, K. J., Barankin, E. W., andBlackwell, D.,Admissible Points of Convex Sets, Contributions to the Theory of Games, Vol. 2, Edited by H. W. Kuhn and A. W. Tucker, Princeton University Press, Princeton, New Jersey, 1953.

    Google Scholar 

  9. Gale, D., Kuhn, H. W., andTucker, A. W.,Linear Programming and the Theory of Games, Activity Analysis of Production and Allocation, Edited by T. C. Koopmans, John Wiley and Sons, New York, New York, 1951.

    Google Scholar 

  10. Kuhn, H. W., andTucker, A. W.,Nonlinear Programming, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, California, 1950.

    Google Scholar 

  11. Koopmans, T. C.,Three Essays on the State of Economic Science, McGraw-Hill Book Company, New York, New York, 1957.

    Google Scholar 

  12. Geoffrion, A. M.,Proper Efficiency and the Theory of Vector Maximization, Journal of Mathematical Analysis and Applications, Vol. 22, pp. 618–630, 1968.

    Google Scholar 

  13. Geoffrion, A. M.,Strictly Concave Parametric Programming, Parts I and II, Management Science, Vol. 13, pp. 244–253 and 359–370, 1967.

    Google Scholar 

  14. Geoffrion, A. M.,Solving Bicriterion Mathematical Programs, Operations Research, Vol. 15, pp. 39–54, 1967.

    Google Scholar 

  15. Yu, P. L.,Cone Convexity, Cone Extreme Points, and Nondominated Solutions in Decision Problems with Multiobjectives, Journal of Optimization Theory and Applications, Vol. 14, pp. 319–377, 1974.

    Google Scholar 

  16. Smale, S.,Global Analysis and Economics, III: Pareto Optima and Price Equilibria, Journal of Mathematical Economics, Vol. 1, pp. 107–117, 1974.

    Google Scholar 

  17. Smale, S.,Global Analysis and Economics, V: Pareto Theory with Constraints, Journal of Mathematical Economics, Vol. 1, pp. 213–221, 1974.

    Google Scholar 

  18. Smale, S.,Global Analysis and Economics, VI: Geometric Analysis of Pareto Optima and Price Equilibria under Classical Hypotheses, Journal of Mathematical Economics, Vol. 3, pp. 1–14, 1976.

    Google Scholar 

  19. Rand, D.,Thresholds in Pareto Sets, Journal of Mathematical Economics, Vol. 3, pp. 139–154, 1976.

    Google Scholar 

  20. Simon, C. P., andTitus, C.,Characterization of Optima in Smooth Pareto Economic Systems, Journal of Mathematical Economics, Vol. 2, pp. 297–330, 1975.

    Google Scholar 

  21. Wan, Y. H.,On Local Pareto Optimum, Journal of Mathematical Economics, Vol. 2, pp. 35–42, 1975.

    Google Scholar 

  22. Charnes, A., andCooper, W. W.,Management Models and Industrial Applications of Linear Programming, Vol. 1, Chapter 9, John Wiley and Sons, New York, New York, 1961.

    Google Scholar 

  23. Ecker, J. G., andKouada, I. A.,Finding Efficient Points for Linear Multiple Objective Programs, Mathematical Programming, Vol. 8, pp. 375–377, 1975.

    Google Scholar 

  24. Ecker, J. G., andKouada, I. A.,Generating Maximal Efficient Faces for Multiple Objective Linear Programs, Université Catholique de Louvain, Heverlee, Belgium, CORE, Discussion Paper No. 7617, 1976.

    Google Scholar 

  25. Evans, J. P., andSteuer, R. E.,A Revised Simplex Method for Linear Multiple Objective Programs, Mathematical Programming, Vol. 5, pp. 54–72, 1973.

    Google Scholar 

  26. Gal, T.,A General Method for Determining the Set of All Efficient Solutions to a Linear Vector Maximum Problem, Institut für Wirstschaftswissenschaften, Aachen, Germany, Report No. 76/12, 1976.

    Google Scholar 

  27. Geoffrion, A. M., Dyer, J. S., andFeinberg, A.,An Interactive Approach for Multi-Criterion Optimization with an Application to the Operation of an Academic Department, Management Science, Vol. 19, pp. 357–368, 1972.

    Google Scholar 

  28. Philip, J.,Algorithms for the Vector Maximization Problem, Mathematical Programming, Vol. 2, pp. 207–229, 1972.

    Google Scholar 

  29. Schachtman, R.,Generation of the Admissible Boundary of a Convex Polytope, Operations Research, Vol. 22, pp. 151–159, 1974.

    Google Scholar 

  30. Yu, P. L., andZeleny, M.,The Set of All Nondominated Solutions in Linear Cases and a Multicriteria Simplex Method, Journal of Mathematical Analysis and Applications, Vol. 49, pp. 430–468, 1975.

    Google Scholar 

  31. Ferguson, T. S.,Mathematical Statistics, Academic Press, New York, New York, 1967.

    Google Scholar 

  32. Whittle, P.,Optimization Under Constraints, Chapter 10, John Wiley and Sons, New York, New York, 1971.

    Google Scholar 

  33. Markowitz, H.,The Optimization of Quadratic Functions Subject to Linear Constraints, Naval Research Logistics Quarterly, Vol. 3, pp. 111–133, 1956.

    Google Scholar 

  34. Markowitz, H.,Portfolio Selection: Efficient Diversification of Investments, John Wiley and Sons, New York, New York, 1962.

    Google Scholar 

  35. Raiffa, H.,Decision Analysis: Introductory Lectures on Choice Under Uncertainty, Addison-Wesley Publishing Company, Reading, Massachusetts, 1970.

    Google Scholar 

  36. Bitran, G. R.,Admissible Points and Vector Optimization: A Unified Approach, Massachusetts Institute of Technology, Operations Research Center, PhD Thesis, 1975.

  37. Cochrane, J. L., andZeleny, M., Editors,Multiple Criteria Decision Making, University of South Carolina Press, Columbus, South Carolina, 1973.

    Google Scholar 

  38. Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.

    Google Scholar 

  39. Stoer, J., andWitzgall, C.,Convexity and Optimization in Finite Dimensions I, Springer-Verlag, Berlin, Germany, 1970.

    Google Scholar 

  40. Berge, C.,Topological Spaces, The MacMillan Company, New York, New York, 1963.

    Google Scholar 

  41. Hildenbrand, W., andKirman, A. P.,Introduction to Equilibrium Analysis, North-Holland Publishing Company, Amsterdam, Holland, 1976.

    Google Scholar 

  42. Mangasarian, O. L.,Nonlinear Programming, McGraw-Hill Book Company, New York, New York, 1969.

    Google Scholar 

  43. Fiacco, A. V., andMcCormick, G. P.,Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley and Sons, New York, New York, 1968.

    Google Scholar 

  44. Magnanti, T. L.,A Linear Approximation Approach to Duality in Nonlinear Programming, Massachusetts Institute of Technology, Operations Research Center, Working Paper No. OR-016-73, 1973.

  45. Halkin, H.,Nonlinear Nonconvex Programming in an Infinite Dimensional Space, Mathematical Theory of Control Edited by A. V. Balakrishnan and L. W. Neustadt, Academic Press, New York, New York, 1967.

    Google Scholar 

  46. Robinson, S. M.,First Order Conditions for General Nonlinear Optimization, SIAM Journal on Applied Mathematics, Vol. 30, pp. 597–607, 1976.

    Google Scholar 

  47. Borwein, J.,Proper Efficient Points for Maximizations with Respect to Cones, SIAM Journal on Control and Optimization, Vol. 15, pp. 57–63, 1977.

    Google Scholar 

  48. Naccache, P.,Stability in Multicriteria Optimization, University of California at Berkeley, Department of Mathematics, PhD Thesis, 1977.

Download references

Author information

Authors and Affiliations

  1. Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts

    G. R. Bitran (Assistant Professor) & T. L. Magnanti (Professor)

Authors
  1. G. R. Bitran
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. L. Magnanti
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by P. Varaiya

Several results from this paper were presented in less general form at the National ORSA/TIMS Meeting, Chicago, Illinois, 1975.

This research was supported, in part, by the United States Army Research Office (Durham), Grant No. DAAG-29-76-C-0064, and by the Office of Naval Research, Grant No. N00014-67-A-0244-0076. The research of the second author was partially conducted at the Center for Operations Research and Econometrics (CORE), Université Catholique de Louvain, Heverlee, Belgium.

The authors are indebted to A. Assad for several helpful discussions and to A. Weiczorek for his careful reading of an earlier version of this paper.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bitran, G.R., Magnanti, T.L. The structure of admissible points with respect to cone dominance. J Optim Theory Appl 29, 573–614 (1979). https://doi.org/10.1007/BF00934453

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00934453

Key Words

Search

Navigation