Epsilon efficiency

  • D. J. White
Contributed Papers

Abstract

This paper considers the extension of ε-optimality for scalar problems to vector maximization problems, or efficiency problems, which havem objective functions defined on a set\(X \subseteq \mathbb{R}^n \).

It is shown that the natural extension of the scalar ε-optimality concepts [viz, given ε>0, given a solution setS, ifx∈S there exists an efficient solutiony with ∥f(x)−f(y)∥≦ε, and given an efficient solutiony, there exists anx∈S with ∥f(x)−f(y)∥≦ε] do not hold for some methods used. Six concepts of ε-efficient sets are introduced and examined, to a very limited extent, in the context of five methods used for generating efficient points or near efficient points.

In doing so, a distinction is drawn between methods in which the surrogate optimizations are carried out exactly, and those where terminal ε-optimal solutions are obtained.

Key Words

Efficient sets ε-efficiency weighting factors constrained objectives penalty functions ideal points Markov decision processes 
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References

  1. 1.
    White, D. J.,Optimality and Efficiency, Wiley, Chichester, England, 1983.Google Scholar
  2. 2.
    White, D. J.,Characterisation of Efficient Sets by Constrained Objectives, Journal of Optimization Theory and Applications, Vol. 45, pp. 603–629, 1985.Google Scholar
  3. 3.
    White, D. J.,Multi-Objective Interactive Programming for Efficient Sets, Multi-Objective Decision Making, Edited by S. French, R. Hartley, L. C. Thomas, and D. J. White, Academic Press, London, England, 1983.Google Scholar
  4. 4.
    White, D. J.,Multi-Objective Programming and Penalty Functions, Journal of Optimization Theory and Applications, Vol. 45, pp. 583–599, 1985.Google Scholar
  5. 5.
    White, D. J.,Multi-Objective, Infinite Horizon, Discounted Markov Decision Processes, Journal of Mathematical Analysis and Applications, Vol. 89, pp. 639–647, 1982.Google Scholar
  6. 6.
    Dinkelbach, W., andIsermann, H.,On Decision Making under Multiple Criteria and under Incomplete Information, Multiple Criteria Decision Making, Edited by J. L. Cochrane and M. Zeleny, University of South Carolina Press, Columbia, South Carolina, 1973.Google Scholar
  7. 7.
    Sawaragi, Y., andTanino, T.,Duality Theory and Multi-Objective Programming, Journal of Optimization Theory and Applications, Vol. 27, pp. 509–530, 1979.Google Scholar
  8. 8.
    Karlin, S.,Mathematical Methods and Theory in Games, Programming, and Economics, Addison-Wesley, Reading, Massechusetts, 1959.Google Scholar
  9. 9.
    Polak, E.,On the Approximation of Solutions to Multiple-Criteria Decision-making Problems, Multiple Criteria Decision Making, Edited by M. Zeleny, Springer-Verlag, Berlin, Germany, 1976.Google Scholar
  10. 10.
    Frank, M., andWolfe, P.,An Algorithm for Quadratic Programming, Naval Research Logistics Quarterly, Vol. 3, pp. 95–110, 1956.Google Scholar
  11. 11.
    Geoffrion, A. M.,Perspectives in Optimization, Addison-Wesley, Reading, Massachusetts, 1972.Google Scholar
  12. 12.
    Zangwill, W. I.,Nonlinear Programming, Prentice-Hall, Englewood Cliffs, New Jersey, 1969.Google Scholar

Copyright information

© Plenum Publishing Corporation 1986

Authors and Affiliations

  • D. J. White
    • 1
  1. 1.University of ManchesterManchesterEngland

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