Three Methods for Determining Pareto-Optimal Solutions of Multiple-Objective Problems

  • Jiguan G. Lin


Typical problems in system design, decision making, decentralized control, etc., and most multi-person games bear the following general formulation of multiple-objective (MO) optimization problems:
$${\text{maximize }}{{\text{z}}_{\text{1}}} = {{\text{J}}_{\text{1}}}\left( {{{\text{x}}_1}, \ldots ,{{\text{x}}_{\text{n}}}} \right), \ldots ,\,\,{\text{and}}\,{{\text{z}}_{\text{N}}} = {{\text{J}}_{\text{N}}}\left( {{{\text{x}}_{\text{1}}}, \ldots ,{{\text{x}}_{\text{n}}}} \right)\,{\text{subject to }}\left( {{{\text{x}}_1}, \ldots ,{{\text{x}}_{\text{n}}}} \right) \in {\text{X}}{\text{.}}$$


Differential Game Positive Weight Positive Vector Equality Constant Maximal Vector 
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. [1]
    L. A. Zadeh, “Optimality and Non-Scalar-Valued Performance Criteria,” IEEE Trans, Automat. Contr., vol. AC-8, pp. 59–60, Jan. 1963.CrossRefGoogle Scholar
  2. [2]
    J. Von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, 3rd ed. Princeton: Princeton Univ. Press, 1953.Google Scholar
  3. [3]
    Y.C. Ho, “Differential Games, Dynamic Optimization, and Generalized Control Theory,” J. Optimiz. Theory Appl. vol. 6, no. 3, pp. 179–209, 1970.Google Scholar
  4. [4]
    J. G. Lin, “On N-Person Cooperative Differential Games,” Proc. 6th Princeton Conf. Inform. Sci. Amp; Syst., Princeton, N.J., pp. 502–507, Mar. 1972.Google Scholar
  5. [5]
    T. C. Koopmans, “Analysis of Production as an Efficient Combination of Activities,” Activity Analysis of Production and Allocation, T. C. Koopmans, Ed. New York: Wiley, pp. 33–97, 1951.Google Scholar
  6. [6]
    K.J. Arrow, E. W. Barankin, and D. Blackwell, “Admissible Points of Convex Sets,” Contributions to the Theory of Games, vol.II, H. W. Kuhn and A. W. Tucker, Eds. Princeton: Princeton Univ. Press, pp. 87–91, 1953.Google Scholar
  7. [7]
    N.O. Da Cuncha and E. Polak, “Constrained Minimization under Vector-Valued Criteria in Finite Dimensional Spaces,” J. Math. Anal. and Appl. vol.19, no.1, pp.103–124, July 1967.Google Scholar
  8. [8]
    J. G. Lin, “Multiple-Objective Optimization,” Columbia Univ., Dept. of Elect. Eng. & Comp. Sci., Syst. Res. Gr. Tech. Rept, Dec. 1972.Google Scholar
  9. [9]
    G. A. Katopis and J. G. Lin, “Non-Inferiority of Controls under Double Performance Objectives: Minimal Time and Minimal Energy,” Proc. 7th Hawaii Int. Conf. Syst. Sci., Honolulu, Hawaii, pp. 129–131, Jan. 1974.Google Scholar
  10. [10]
    J. G. Lin, “Circuit Design under Multiple Performance Objectives,” Proc. 1974 IEEE Int. Symp. Circuits & Systems, San Francisco, Calif., pp. 549–552, Apr. 1974.Google Scholar
  11. [11]
    J.G. Lin, “Maximal Vectors and Multi-Objective Optimization,” J. Optimiz. Theory Appl., vol.18, no.1, pp.41–64, Jan. 1976.Google Scholar
  12. [12]
    J.G. Lin, “Proper Equality Constraints (PEC) and Maximization of Index Vectors,” J. Optimiz. Theory Appl. vol.20, no.4, Dec.1976.Google Scholar
  13. [13]
    J.G. Lin, “Proper Inequality Constraints (PIC) and Maximization of Index Vectors” to appear in J. Optimiz. Theory Appl..Google Scholar
  14. [14]
    J.G. Lin, “Multiple-Objective Problems: Pareto-Optimal Solutions by Method of Proper Equality Constraints (PEC),” to appear in IEEE Trans. Automatic Control.Google Scholar
  15. [15]
    J. G. Lin, “Multiple-Objective Programming: Lagrange Multipliers and Method of Proper Equality Constraints,” to be presented in 1976 Joint Automatic Control Conf., July 1976.Google Scholar
  16. [16]
    P.L. Yu, “Cone Convexity, Cone Extreme Points, and Nondominated Solutions in Decision Problems with Multi-objectives,” J. Optimiz. Theory Appl. vol.14, no.3, pp.319–377, Sept. 1974.Google Scholar
  17. [17]
    H. Payne, E. Polak, D. C. Collins, and W.S. Meisel, “An Algorithm for Bicriteria Optimization Based on the Sensitivity Function.” IEEE Trans. Automat. Contr. vol. AC-20, no. 4, pp. 546–548, Aug. 1975.CrossRefGoogle Scholar
  18. [18]
    J. M. Holtzman and H. Halkin, “Directional Convexity and the Maximum Principle for Discrete Systems,” J. SIAM Contr., vol.4, no. Z, pp. Z63–Z75, 1966.Google Scholar
  19. [19]
    D. M. Himmelblau, Applied Nonlinear Programming, New York: McGraw-Hill, 1972.Google Scholar
  20. [20]
    A. E. Bryson and Y. C. Ho, Applied Optimal Control, Waltham, Mass.: Blaisdell, 1969.Google Scholar

Copyright information

© Plenum Press, New York 1976

Authors and Affiliations

  • Jiguan G. Lin
    • 1
  1. 1.Department of Electrical Engineering & Comp. Sci.Columbia UniversityNew YorkUSA

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