Introduction to Convex and Quasiconvex Analysis

  • Johannes B.G. Frenk
  • Gábor Kassay
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 76)

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 concerning quasiconvex-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.

Keywords

Convex Analysis Quasiconvex Analysis Noncooperative games Minimax Optimization theory 

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References

  1. [1]
    Aubin, J.B., Optima and Equilibria (An introduction to nonlinear analysis), Graduate Texts in Mathematics, v. 140, Springer Verlag, Berlin, 1993.Google Scholar
  2. [2]
    Avriel, M., Diewert, W.E., Schaible, S. and I. Zang, Generalized Concavity, in Mathematical Concepts and Methods in Science and Engineering, v. 36, Plenum Press, New York, 1988.Google Scholar
  3. [3]
    Bazaara, M.S., Sherali, H.D. and C.N. Shetty, Nonlinear Programming (Theory and Applications), Wiley, New York, 1993.Google Scholar
  4. [4]
    Bazaara, M.S., Jarvis, J.J. and H.D. Sherali, Linear Programming and Network Flows (Second Edition), Wiley, New York, 1990.Google Scholar
  5. [5]
    Bonnesen, T. and W. Fenchel, Theorie der konvexen Körper, Springer Verlag, Berlin, 1974.Google Scholar
  6. [6]
    Breckner, W.W. and G. Kassay, A systematization of convexity concepts for sets and functions, Journal of Convex Analysis, 4, 1997, 1–19.MathSciNetGoogle Scholar
  7. [7]
    Brunn, H., Über Ovale und Eiflächen, Inauguraldissertation, University of München, 1887.Google Scholar
  8. [8]
    Brunn, H., Über Kurven ohne Wendepunkte, Habilitationsschrift, University of München, 1889.Google Scholar
  9. [9]
    Brunn, H., Zur Theorie der Eigebiete, Arch. Math. Phys. Ser. 3, 17, 1910, 289–300.Google Scholar
  10. [10]
    Choquet, G., Lectures on Analysis: Volume 2 Representation Theory, in Mathematics Lecture Note Series, W.A. Benjamin, London, 1976.Google Scholar
  11. [11]
    Crouzeix, J.-P., Continuity and differentiability of quasiconvex functions, Chapter 3 in this volume.Google Scholar
  12. [12]
    Crouzeix, J.-P., Contributions a l’ étude des fonctions quasiconvexes, Université de Clermond-Ferrand 2, France, 1977.Google Scholar
  13. [13]
    Crouzeix, J.-P., Conditions for convexity of quasiconvex functions, Mathematics of Operations Research, 5, 1980, 120–125.MATHMathSciNetGoogle Scholar
  14. [14]
    Crouzeix, J.-P., A duality framework in quasiconvex programming, in Generalized Concavity in Optimization and Economics, Schaible, S. and W.T. Ziemba, eds., Academic Press, 1981, 207–225.Google Scholar
  15. [15]
    Crouzeix, J.-P., Continuity and differentiability properties of quasiconvex functions on ℝn, in Generalized Concavity in Optimization and Economics, Schaible, S. and W.T. Ziemba, eds., Academic Press, 1981, 109–130.Google Scholar
  16. [16]
    De Finetti, B., Sulle stratificazioni convesse, Ann. Mat. Pura Appl., [4]30, 1949, 173–183.Google Scholar
  17. [17]
    Dudley, R.M., Real Analysis and Probability, Wadsworth and Brooks/Cole, Pacific Grove, 1989.Google Scholar
  18. [18]
    Edwards, R.E., Functional Analysis: Theory and Applications, Holt, Rinehart and Winston, Chicago, 1965.Google Scholar
  19. [19]
    Faigle, U., Kern, W. and G. Still, Algorithmic Principles of Mathematical Programming, in Kluwer texts in Mathematical Sciences, v. 24, Kluwer Academic Publishers, Dordrecht, 2002.Google Scholar
  20. [20]
    Farkas, J., Uber die Theorie der einfache Ungleichungen, Journal für die Reine und Angewandte Mathematik, 124, 1901, 1–27.MATHGoogle Scholar
  21. [21]
    Fenchel, W., On conjugate convex functions, Canadian Journal of Mathematics, 1, 1949, 73–77.MATHMathSciNetGoogle Scholar
  22. [22]
    Fenchel, W., Convex Cones, Sets and Functions, Lecture notes, Princeton University, 1951.Google Scholar
  23. [23]
    Fenchel, W., A remark on convex sets and polarity, Communication Seminar on Mathematics, University of Lund supplementary volume, 1952, 22–89.Google Scholar
  24. [24]
    Frenk, J.B.G and G. Kassay, On classes of generalized convex functions, Gordan-Farkas type theorems and Lagrangian duality, Journal of Optimization Theory and Applications, [2]102, 1999, 315–343.MathSciNetGoogle Scholar
  25. [25]
    Frenk, J.B.G. and G. Kassay, Lagrangian duality and cone convexlike functions, Econometric Institute, Erasmus University Rotterdam, Tech. Rep. 2000-27/A, 2000.Google Scholar
  26. [26]
    Frenk, J.B.G. and G. Kassay, Minimax results and finite dimensional separation, Journal of Optimization Theory and Applications, [2]113, 2002, 409–421.MathSciNetGoogle Scholar
  27. [27]
    Frenk, J.B.G., Dias, D.M.L. and J. Gromicho, Duality theory for convex and quasiconvex Functions and its application to optimization, in Generalized Convexity: Proceedings Pecs, Hungary, 1992, Komlósi, S., Rapcsák, T. and S. Schaible, eds., Lecture notes in Economics and Mathematics, 405, 1994, 153–171.Google Scholar
  28. [28]
    Frenk, J.B.G., Kassay, G. and J. Kolumbán, On equivalent results in minimax theory, European Journal of Operational Research 157, 2004, 46–58.CrossRefMathSciNetGoogle Scholar
  29. [29]
    Frenk, J.B.G., Protassov, V. and G. Kassay, On Borel probability measures and noncooperative game theory, Report Series Econometric Institute, Erasmus University Rotterdam, Tech. Rep. EI-2002-32, 2002.Google Scholar
  30. [30]
    Giorgio, G. and S. Komlósi, Dini derivatives in optimization-Part I, Rivista di Matematica per le Scienze Economiche e Sociali, [1]15, 1992, 3–30.Google Scholar
  31. [31]
    Glover, F., Surrogate constraints, Operations Research, 16, 1967, 741–749.MathSciNetGoogle Scholar
  32. [32]
    Greenberg, H.J. and W.P. Pierskalla, Quasiconjugate functions and surrogate duality, Cahiers du Centre d’étude de Recherche Operationelle, [4]15, 1973, 437–448.MathSciNetGoogle Scholar
  33. [33]
    Gromicho, J., Quasiconvex Optimization and Location Theory, in Applied Optimization, v. 9, Kluwer Academic Publishers, Dordrecht, 1998.Google Scholar
  34. [34]
    Hiriart-Urruty, J.B. and C. Lemaréchal, Convex Analysis and Minimization Algorithms I, in Grundlehren der Mathematischen Wissenschaften, v. 305, Springer Verlag, Berlin, 1993.Google Scholar
  35. [35]
    Hiriart-Urruty, J.B. and C. Lemaréchal, Convex Analysis and Minimization Algorithms II, in Grundlehren der Mathematischen Wissenschaften, v. 306, Springer Verlag, Berlin, 1993.Google Scholar
  36. [36]
    Hocking, J.G and G.S. Young, Topology, Addison Wesley, Reading, Massachusetts, 1961.Google Scholar
  37. [37]
    Holmes, R.B., Geometric Functional Analysis and its Applications, Springer, New York, 1975.Google Scholar
  38. [38]
    Jahn, J., Mathematical Vector Optimization in Partially Ordered Spaces, Peter Lang, Fankfurt am Main, 1986.Google Scholar
  39. [39]
    Jensen, J.L.W.V., Sur les functions convexes et les inégalités entre des valeurs moyennes, Acta Math., 30, 1906, 175–193.MATHGoogle Scholar
  40. [40]
    Joó, I., A simple proof for von Neumann’s minimax theorem, Acta Sci. Math. Szeged, 42, 1980, 91–94.MATHMathSciNetGoogle Scholar
  41. [41]
    Joó, I., Note on my paper: A simple proof for von Neumann’s minimax theorem, Acta Math. Hung., [3–4]44, 1984, 171–176.MathSciNetGoogle Scholar
  42. [42]
    Joó, I., On some convexities, Acta Math.Hung., [1–2]54, 1989, 163–172.Google Scholar
  43. [43]
    Kolmogorov, A.N. and S.V. Fomin, Introductory Real Analysis, in Dover books on Mathematics, Dover Publications, New York, 1975.Google Scholar
  44. [44]
    Komlósi, S., Quasiconvex first-order approximations and Kuhn-Tucker type optimality conditions, European Journal of Operational Research, 65, 1993, 327–335.MATHGoogle Scholar
  45. [45]
    Komlósi, S., Farkas theorems for positively homogeneous quasiconvex functions, Journal of Statistics and Management Systems, 5, 2002, 107–123.MATHMathSciNetGoogle Scholar
  46. [46]
    Kreyszig, E., Introductory Functional Analysis with Applications, Wiley, New York, 1978.Google Scholar
  47. [47]
    Lancaster, P. and M. Tismenetsky, The Theory of Matrices (Second Edition with Applications), in Computer Science and Applied Mathematics, Academic Press, Orlando, 1985.Google Scholar
  48. [48]
    Mandelbrojt, S., Sur les fonctions convexes, C.R. Acad. Sci. Paris, 209, 1939, 977–978.Google Scholar
  49. [49]
    Martínez-Legaz, J.-E.,Generalized convex duality and its economic applications, Chapter 6 in this volume.Google Scholar
  50. [50]
    Martínez-Legaz, J.-E.,Un concepto generalizado deconjugacion, application a las funciones quasiconvexas, M.Sc. Thesis, Universidad de Barcelona, 1981.Google Scholar
  51. [51]
    Minkowski, H., Algemeine Lehrsätze über die convexen Polyeder, Nachr. Ges. Wiss. Göttingen Math. Phys. K1, 1897, 198–219.Google Scholar
  52. [52]
    Minkowski, H., Geometrie der Zahlen, Teubner, Leipzig, 1910.Google Scholar
  53. [53]
    Nesterov, Y. and A. Nemirovski, Interior Point Polynomial Algorithms in Convex Programming, in SIAM Studies in Applied Mathematics, SIAM, Philadelphia, 1994.Google Scholar
  54. [54]
    Nocedal, J. and S.J. Wright, Numerical Optimization, in Springer series in Operations Research, Springer Verlag, New York, 1999.Google Scholar
  55. [55]
    Passy, U. and E.Z. Prisman, Conjugacy in quasi-convex programming, Mathematical Programming, 30, 1984, 121–146.MathSciNetGoogle Scholar
  56. [56]
    Penot, J.-P. and M. Volle, On quasi-convex duality, Mathematics of Operations Research, [4]15, 1990, 597–624.MathSciNetGoogle Scholar
  57. [57]
    Peterson, E.L., The fundamental relations between geometric programming duality, parametric programming duality and ordinary Lagrangian duality, Annals of Operations Research, 105, 2001, 109–153.MATHMathSciNetGoogle Scholar
  58. [58]
    Ponstein, J., Approaches to the Theory of Optimization, Cambridge University Press, Cambridge, 1980.Google Scholar
  59. [59]
    Popoviciu, T., Deux remarques sur les fonctions convexes, Bull. Sc. Acad. Roumaine, 20, 1938, 45–49.MATHGoogle Scholar
  60. [60]
    Popoviciu, T., Les Fonctions Convexes, Hermann, Paris, 1945.Google Scholar
  61. [61]
    Prekopa, A., On the development of optimization theory, American Mathematical Monthly, [7]87, 1980, 527–542.MathSciNetGoogle Scholar
  62. [62]
    Roberts, A.V. and D.E. Varberg, Convex Functions, Academic Press, New York, 1973.Google Scholar
  63. [63]
    Rockafellar, R.T., Convex Analysis, in Princeton Mathematical Series, v. 28, Princeton University Press, Princeton, New Jersey, 1972.Google Scholar
  64. [64]
    Rudin, W., Principles of Mathematical Analysis, McGraw-Hill, New York, 1976.Google Scholar
  65. [65]
    Rudin, W., Functional Analysis, McGraw-Hill, New York, 1991.Google Scholar
  66. [66]
    Sion, M., On general minimax theorems, Pacific Journal of Mathematics, 8, 1958, 171–176.MATHMathSciNetGoogle Scholar
  67. [67]
    Sturm, J.F., Primal-Dual Interior Point Approach to Semidefinite Programming, PhD. Thesis, Econometric Institute, Erasmus University Rotterdam, 1997.Google Scholar
  68. [68]
    Valentine, F.A., Convex Sets, Mc-Graw Hill, San Francisco, 1964.Google Scholar
  69. [69]
    van Tiel, J., Convex Analysis: an Introductory Text, Wiley, New York, 1984.Google Scholar
  70. [70]
    Ville, J., Sur la théorie générale des jeux au intervient l’habilité des jouers, [2](5), in Traité du calcul des probabilités et de ses applications, E. Borel and others, eds., Gauthier-Villars Cie, Paris, 1938, 105–113.Google Scholar
  71. [71]
    Von Neumann, J., Zur Theorie der Gesellschaftsspiele, Math. Ann, 100, 1928, 295–320.CrossRefMATHMathSciNetGoogle Scholar
  72. [72]
    Vorob’ev, N.N., Game Theory: Lectures for Economists and Systems Scientists, in Applications of Mathematics, v. 7, Springer Verlag, New York, 1977.Google Scholar
  73. [73]
    Wolkowicz, H., Some applications of optimization in matrix theory, Linear Algebra and Its Applications, 40, 1981, 101–118.CrossRefMATHMathSciNetGoogle Scholar
  74. [74]
    Yuang, G.X.-Z., KKM Theory and Applications in Nonlinear Analysis, Marcel Dekker, New York, 1999.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • Johannes B.G. Frenk
    • 1
  • Gábor Kassay
    • 2
  1. 1.Department of Economics, Econometric InstituteErasmus UniversityRotterdamThe Netherlands
  2. 2.Faculty of Mathematics and Computer ScienceBabes Bolyai UniversityClujRomania

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