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Generalized Homotopy Approach to Multiobjective Optimization

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This paper proposes a new generalized homotopy algorithm for the solution of multiobjective optimization problems with equality constraints. We consider the set of Pareto candidates as a differentiable manifold and construct a local chart which is fitted to the local geometry of this Pareto manifold. New Pareto candidates are generated by evaluating the local chart numerically. The method is capable of solving multiobjective optimization problems with an arbitrary number k of objectives, makes it possible to generate all types of Pareto optimal solutions, and is able to produce a homogeneous discretization of the Pareto set. The paper gives a necessary and sufficient condition for the set of Pareto candidates to form a (k-1)-dimensional differentiable manifold, provides the numerical details of the proposed algorithm, and applies the method to two multiobjective sample problems.

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  1. Stadler, W., Editor, Multicriteria Optimization in Engineering and in the Sciences, Plenum Press, New York, NY, 1988.

    Google Scholar 

  2. Zadeh, L., Optimality and Nonscalar-Valued Performance Criteria, IEEE Transactions on Automatic Control, Vol. 8, pp. 59–60, 1963.

    Google Scholar 

  3. Haimes, Y. Y., Integrated System Identification and Optimization, Control and Dynamic Systems: Advances in Theory and Applications, Edited by C. T. Leondes, Academic Press, New York, NY, Vol. 10, pp. 435–518, 1973.

    Google Scholar 

  4. Das, I., and Dennis, J., Normal-Boundary Intersection: A New Method for Generating Pareto-Optimal Points in Multicriteria Optimization Problems, Technical Report96–11, Department of Computational and Applied Mathematics, Rice University, Houston, Texas, 1996.

    Google Scholar 

  5. Das, I., Nonlinear Multicriteria Optimization and Robust Optimality, PhD Thesis, Rice University, Houston, Texas, 1997.

    Google Scholar 

  6. Rakowska, J., Haftka, R. T., and Watson, L. T., Tracing the Efficient Curve for Multiobjective Control-Structure Optimization, Computing Systems in Engineering, Vol. 2, pp. 461–471, 1991.

    Google Scholar 

  7. Hillermeier, C., Eine Homotopiemethode zur Vektoroptimierung, Habilitation Thesis, Technical University, München, Germany, 1999.

    Google Scholar 

  8. Kuhn, H., and Tucker, A., Nonlinear Programming, Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, Edited by J. Neyman, University of California, Berkeley, California, pp. 481–492, 1951.

    Google Scholar 

  9. Luenberger, D. G., Linear and Nonlinear Programming, Addison-Wesley Publishing Company, Reading, Massachusetts, 1984.

    Google Scholar 

  10. Fletcher, R., Practical Methods of Optimization, John Wiley, New York, NY, 1987.

    Google Scholar 

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Hillermeier, C. Generalized Homotopy Approach to Multiobjective Optimization. Journal of Optimization Theory and Applications 110, 557–583 (2001).

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