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A Survey on Separability and Generalized Convexity or Generalized Monotonicity

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Recent Developments in Optimization

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 429))

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Abstract

In this paper, we give necessary and sufficient conditions for a separable sum of functions to be quasiconvex, a separable product of positive functions to be convex, concave, quasiconvex, quasiconcave, a separable product of operators to be quasimonotone. These conditions are expressed in terms of convexity indices or monotonicity indices.

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References

  1. J.P. Crouzeix and P.O. Lindberg, “Additively decomposed quasiconvex functions”, Mathematical Programming 27 (1986) 42–57.

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  2. J.P. Crouzeix and R. Kebbour, “Index multiplicatifs de convexité/concavité et applications”, Cahiers du Centre d’Études de Recherche Opérationnelle 34 (1992) 7–23.

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  3. J.P. Crouzeix and A. Hassouni, “Quasimonotonicity of separable operators and monotonicity indices”, SIAM Journal of Optimization 4 (1994) 649–658.

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  4. J.P. Crouzeix and A. Hassouni, “Generalized Monotonicity of a separable product of operators: the multivalued case”, Working Paper, Université Blaise Pascal (Clermont-Ferrand, 1993 ).

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  5. G. Debreu G. and T.C. Koopmans, “Additively decomposed quasiconvex functions”, Mathematical Programming 24 (1982) 1–38.

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  6. M.E. Yaari, “A note on separability and quasiconvexity”, Econometrica 45 (1977) 1183–1186.

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Author information

Authors and Affiliations

  1. Mathématiques Appliquées, Université Blaise Pascal, URA 1501 CNRS, 63177, Aubière Cedex, France

    Jean-Pierre Crouzeix

Authors
  1. Jean-Pierre Crouzeix
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Editor information

Editors and Affiliations

  1. Analyse Appliquée et Optimisation, University of Bourgogne, P. O. 138, F-21004, Dijon Cedex, France

    Roland Durier  & Christian Michelot  & 

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© 1995 Springer-Verlag Berlin Heidelberg

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Crouzeix, JP. (1995). A Survey on Separability and Generalized Convexity or Generalized Monotonicity. In: Durier, R., Michelot, C. (eds) Recent Developments in Optimization. Lecture Notes in Economics and Mathematical Systems, vol 429. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-46823-0_9

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  • DOI: https://doi.org/10.1007/978-3-642-46823-0_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60041-1

  • Online ISBN: 978-3-642-46823-0

  • eBook Packages: Springer Book Archive

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