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Introduction to Convex and Quasiconvex Analysis

Author

Listed:
  • Frenk, J.B.G.
  • Kassay, G.
Abstract
In the first chapter of this book the basic results within convex and quasiconvex analysis are presented. In Section 2 we consider in detail the algebraic and topological properties of convex sets within Rn together with their primal and dual representations. In Section 3 we apply the results for convex sets to convex and quasiconvex functions and show how these results can be used to give primal and dual representations of the functions considered in this field. As such, most of the results are well-known with the exception of Subsection 3.4 dealing with dual representations of quasiconvex functions. In Section 3 we consider applications of convex analysis to noncooperative game and minimax theory, Lagrangian duality in optimization and the properties of positively homogeneous evenly quasiconvex functions. Among these result an elementary proof of the well-known Sion’s minimax theorem concerningquasiconvex-quasiconcave bifunctions is presented, thereby avoiding the less elementary fixed point arguments. Most of the results are proved in detail and the authors have tried to make these proofs as transparent as possible. Remember that convex analysis deals with the study of convex cones and convex sets and these objects are generalizations of linear subspaces and affine sets, thereby extending the field of linear algebra. Although some of the proofs are technical, it is possible to give a clear geometrical interpretation of the main ideas of convex analysis. Finally in Section 5 we list a short and probably incomplete overview on the history of convex and quasiconvex analysis.

Suggested Citation

  • Frenk, J.B.G. & Kassay, G., 2004. "Introduction to Convex and Quasiconvex Analysis," ERIM Report Series Research in Management ERS-2004-075-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    • Handle: RePEc:ems:eureri:1611
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    File URL: https://repub.eur.nl/pub/1611/ERS%202004%20075%20LIS.pdf
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    References listed on IDEAS

    as
    1. Frenk, J.B.G. & Kassay, G. & Protassov, V., 2002. "On Borel Probability Measures and Noncooperative Game Theory," ERIM Report Series Research in Management ERS-2002-85-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    2. J. B. G. Frenk & G. Kassay, 1999. "On Classes of Generalized Convex Functions, Gordan–Farkas Type Theorems, and Lagrangian Duality," Journal of Optimization Theory and Applications, Springer, vol. 102(2), pages 315-343, August.
    3. Frenk, J.B.G. & Kassay, G. & Protassov, V., 2002. "On Borel Probability Measures and Noncooperative Game Theory," Econometric Institute Research Papers ERS-2002-85-LIS, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    4. Elmor Peterson, 2001. "The Fundamental Relations between Geometric Programming Duality, Parametric Programming Duality, and Ordinary Lagrangian Duality," Annals of Operations Research, Springer, vol. 105(1), pages 109-153, July.
    5. J. P. Crouzeix, 1980. "Conditions for Convexity of Quasiconvex Functions," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 120-125, February.
    6. Jean-Paul Penot & Michel Volle, 1990. "On Quasi-Convex Duality," Mathematics of Operations Research, INFORMS, vol. 15(4), pages 597-625, November.
    7. J.B.G. Frenk & G. Kassay & V. Protassov, 2002. "On Borel Probability Measures and Noncooperative Game Theory," Tinbergen Institute Discussion Papers 02-093/4, Tinbergen Institute.
    8. Frenk, J. B. G. & Kassay, G. & Kolumban, J., 2004. "On equivalent results in minimax theory," European Journal of Operational Research, Elsevier, vol. 157(1), pages 46-58, August.
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    Cited by:

    1. Frenk, J.B.G. & Still, G.J., 2005. "A Note on the Dual of an Unconstrained (Generalized) Geometric Programming Problem," ERIM Report Series Research in Management ERS-2005-006-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.

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    More about this item

    Keywords

    convex analysis; lagrangian dual; minimax theorems; noncooperative games; optimalisatie; optimization theory; quasiconvex analysis;
    All these keywords.

    JEL classification:

    • C69 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Other
    • M - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics
    • M11 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Business Administration - - - Production Management
    • R4 - Urban, Rural, Regional, Real Estate, and Transportation Economics - - Transportation Economics

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