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Introducing oblique norms into multiple criteria programming

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Abstract

We propose to use block norms to generate nondominated solutions of multiple criteria programs and introduce the new concept of the oblique norm that is specially tailored to handle general problems. We prove the equivalence of finding the properly nondominated solutions of a multiple criteria program and solving its scalarization by means of oblique norms.

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References

  • Anderson, D. H. and Osborne, M. R. (1977). Discrete, nonlinear approximation problems in polyhedral norms. Numerische Mathematik, 28, 143–156.

    Google Scholar 

  • Bauer, F. L., Stoer, J. and Witzgall, C. (1961), Absolute and monotonic norms. Numerische Mathematik, 3, 257–264.

    Google Scholar 

  • Carrizosa, E., Conce, E., Pascual, A. and Romero-Morales, D. (1997), Closest solutions in ideal-point methods. In Advances in Multiple Objective and Goal Programming (edited by R. Caballero, F. Ruiz and R. E. Steuer), pp. 274–281. Springer, Berlin.

    Google Scholar 

  • Dubov, Yu. A. (1981), Stability of Pareto-optimal vector evaluations and &3x2208;-uniform solutions. Automation and Remote Control42, 815–821.

    Google Scholar 

  • Ester, J. (1986). Concepts of efficiency and fuzzy aggregation rules. In Large-Scale Modelling and Interactive Decision Analysis (edited by G. Fandel, M. Grauer, A. Kurzhanski and A. B. Wierzbicki), Volume 273 of Lecture Notes in Economics and Mathematical Systems, pp. 59–66. Springer, Berlin.

    Google Scholar 

  • Gearhart, W. B. (1979), Compromise solutions and estimation of the noninferior set. Journal of Optimization Theory and Applications 28, 29–47.

    Google Scholar 

  • Geoffrion, A. M. (1968) Proper efficiency and the theory of vector maximization. Journal of Mathematical Analysis and Applications 22(3), 618–630.

    Google Scholar 

  • Germeier, Ju. B. (1971). Einführung in die Theorie der Unternehmensforschung (Vvedenie v teoriju issledovanija operacii, Russian). Nauka, Moscow. Hauptredaktion für physikalischmathematische Literatur, 384 S. R. 1.55.

    Google Scholar 

  • Gerth, C. and Weidner, P. (1990), Nonconvex separation theorems and some applications in vector optimization. Journal of Optimization Theory and Application, 67(2), 297–320.

    Google Scholar 

  • Henig, M. I. (1982), Proper efficiency with respect to cones. Journal of Optimization Theory and Applications 36(3), 387–407.

    Google Scholar 

  • Hiriart-Urruty, J. B. and Lemaréchal, C. (1993a), Convex Analysis and Minimization Algorithms I. Springer, Berlin.

    Google Scholar 

  • Hiriart-Urruty, J.B. and Lemaréchal, C. (1993b), Convex Analysis and Minimization Algorithms II. Springer-Verlag, Berlin.

    Google Scholar 

  • Kaliszewski, I. (1987). A modified weighted Tchebycheff metric for multiple objective programming. Computers and Operations Research14, 315–323.

    Google Scholar 

  • Kaliszewski, I. (1994), Quantitative Pareto Analysis by Cone Separation Technique. Kluwer Academic Publishers, Dordrecht.

    Google Scholar 

  • Lewandowski, A. and Wierzbicki, A. (1988), Aspiration based decision analysis, Part I: Theoretical and methodological backgrounds. Working Paper WP-88-03, International Institute for Applied Systems Analysis, Laxenburg, Austria.

  • Minkowski, H. (1967), Gesammelte Abhandlungen, Band 2. Chelsea Publishing Company, New York.

    Google Scholar 

  • Rockafellar, R. T. (1970), Convex Analysis. Princeton University Press, Princeton, NJ.

    Google Scholar 

  • Sawaragi, Y., Nakayama, H. and Tanino, T. (1985), Theory of Multiobjective Optimization, volume 176 of Mathematics in Science and Engineering. Academic Press, Orlando, FL.

    Google Scholar 

  • Schandl, B. (1998). On some properties of gauges. Technical Report 662, Department of Mathematical Sciences, Clemson University, Clemson, SC. Available at http://www.math.clemson.edu/affordability/publications.html (15.12.1999).

    Google Scholar 

  • Schandl, B. (1999), Norm-Based Evaluation and Approximation in Multicriteria Programming. Ph. D. thesis, Clemson University, Clemson, SC. Available at http://www.math.clemson.edu/affordability/publications.html (15.12.1999).

    Google Scholar 

  • Schandl, B., Klamroth, K. and Wiecek, M. M. (2001), Norm-based approximation in bicriteria programming. Computational Optimization and Applications, 20, 23–42.

    Google Scholar 

  • Schandl, B., Klamroth, K. and Wiecek, M. M. (2000), Using block norms in bicriteria optimization. In Research and Practice in Multiple Criteria Decision Making (edited by Y.Y. Haimes and R. E. Steuer), pp. 149–160 Springer, Berlin.

    Google Scholar 

  • Schönfeld, P. (1970), Some duality theorems for the non-linear vectormaximum problem. Unternehmensforschung, 14(1), 51–63.

    Google Scholar 

  • Steuer, R. E. (1986), Multiple Criteria Optimization: Theory, Computation, and Application. Wiley, New York.

    Google Scholar 

  • Steuer, R. E. and Choo, E. U. (1983), An interactive weighted Tchebycheff procedure for multiple objective programming. Mathematical Programming, 26, 326–344.

    Google Scholar 

  • Tuy, H. (1998), Convex Analysis and Global Optimization, volume 22 of Nonconvex Optimization and Its Applications. Kluwer Academic Publishers, Norwell, MA.

    Google Scholar 

  • Ward, J. E. and Wendell, R. E. (1985), Using block norms for location modeling. Operations Research, 33, 1074–1090.

    Google Scholar 

  • Yu, P. L. (1973), A class of solutions for group decision problems. Management Science 19, 936–946.

    Google Scholar 

  • Zeleny, M. (1973), Compromise programming. In Multiple Criteria Decision Making (edited by J.L. Cochrane and M. Zeleny), pp. 262–301. University of South Carolina, Columbia, SC.

    Google Scholar 

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Correspondence to Margaret M. Wiecek.

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Schandl, B., Klamroth, K. & Wiecek, M.M. Introducing oblique norms into multiple criteria programming. Journal of Global Optimization 23, 81–97 (2002). https://doi.org/10.1023/A:1014021806919

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