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On Characterizations of Proper Efficiency for Nonconvex Multiobjective Optimization

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Abstract

In this paper, nonconvex multiobjective optimization problems are studied. New characterizations of a properly efficient solution in the sense of Geoffrion's are established in terms of the stability of one scalar optimization problem and the existence of an exact penalty function of a scalar constrained program, respectively. One of the characterizations is applied to derive necessary conditions for a properly efficient control-parameter pair of a nonconvex multiobjective discrete optimal control problem with linear constraints.

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References

  1. Benson, H. P. and Morin, T. L. (1977), The vector maximization problem: proper efficiency and stability, SIAM Journal on Applied Mathematics 32, 64–72.

    Google Scholar 

  2. Borwein, J. M. (1977), Proper efficient points for maximization with respect to cones, SIAM Journal on Control and Optimization 15, 57–63.

    Google Scholar 

  3. Choo, E. U. and Atkins, D. R. (1983), Proper Efficiency in Nonconvex Multicriteria Programming, Mathematics of Operations Research 8, 467–470.

    Google Scholar 

  4. Clarke, F. H. (1983), Optimization and Nonsmooth Analysis, John Wiley & Sons, New York.

    Google Scholar 

  5. Deng, S. (1998), on efficient solutions in vector optimization, Journal of Optimization Theory and Applications 96, 201–209.

    Google Scholar 

  6. Geffrion, A. M. (1967), Solving bicritrion mathematical programs, Operations Research 15, 39–45.

    Google Scholar 

  7. Hiriart-Urruty J. B. and Lemmarechal, C. (1993), Convex Analysis and Minimization Algorithms I, Springer, Berlin.

    Google Scholar 

  8. Jahn, J. (1985), A characterization of properly minimal elements of a set, SIAM Journal on Control and Optimization 23, 649–656.

    Google Scholar 

  9. Jennings, L. S., Fischer, M. E., Teo, K. L. and Goh, C. J. (1990), MISER3, Optimal Control Software, Theory and User Manual, EMCOSS Pty Ltd, 7 Topaz Place, Carine, WA 6020, Australia.

  10. Li, D. (1990), On the minimax solution of multiple linear quadratic problems, IEEE Transactions on Automatic Control 35, 1153–1156.

    Google Scholar 

  11. Li, D. (1993), On general multiple linear quadratic control problems, IEEE Transactions on Automatic Control 38, 1722–1726.

    Google Scholar 

  12. Liao, L. and Li, D. (2000), Successive method for general multiple linear quadratic control problems in discrete time, IEEE Transactions on Automatic Control 45, 1380–1384.

    Google Scholar 

  13. Liu, L. P. (1993), Characterization of nondominated controls in terms of solutions of weighting problems, Journal of Optimization Theory and Applications 77, 545–561.

    Google Scholar 

  14. Rosenberg, E. (1984), Exact penalty functions and stability in locally Lipschitz programming, Mathematical Programming 30, 340–356.

    Google Scholar 

  15. Salukvadze, M. (1974), On the existence of solutions in problems of optimization under vector-valued criteria, Journal of Optimization Theory and Applications 13, 203–217.

    Google Scholar 

  16. Sawaragi, Y., Nakayama, H. and Tanino, T. (1985), Theory of Multiobjective Optimization, Academic Press, New York.

    Google Scholar 

  17. Teo, K. L., Goh, C. J. and Wong, K. H. (1990), A Unified Computational Approach to Optimal Control Problems, Logman Scientific and Technical, New York.

    Google Scholar 

  18. Toivonen, H. T. (1984), A multiobjective linear quadratic Gaussian control problem, IEEE Transactions on Automatic Control 29, 279–280.

    Google Scholar 

  19. Yang, X. Q. and Teo, K. L. (1999), Necessary optimality conditions for bicriterion discrete time optimal control problems, Journal of the Australian Mathematical Society, Ser. B 40, 392–402.

    Google Scholar 

  20. Yu, P. L. and Leitmann, G. (1974), Nondominated decision and cone convexity in dynamic multicriteria decision problems, Journal of Optimization Theory and Applications 14, 573–584.

    Google Scholar 

  21. Zadeh, L. A. (1963), Optimality and non-scalar-valued performance criteria, IEEE Transactions on Automatic Control 8, 59–60.

    Google Scholar 

  22. Zalmai, G. L. (1996), Proper efficiency and duality for a class of constrained multiobjective fractional optimal control problems containing arbitrary norms, Journal of Optimization Theory and Applications 90, 435–456.

    Google Scholar 

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Author notes

  1. X.X. Huang

    Present address: Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Chongqing Normal University, Chongqing, 400047, China

    X.X. Huang

  2. Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong

    X.Q. Yang

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  1. X.X. Huang
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  2. X.Q. Yang
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Correspondence to X.Q. Yang.

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Huang, X., Yang, X. On Characterizations of Proper Efficiency for Nonconvex Multiobjective Optimization. Journal of Global Optimization 23, 213–231 (2002). https://doi.org/10.1023/A:1016522528364

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