Convex Analysis

  • Robert J. Vanderbei
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 37)

Abstract

This book is mostly about linear programming. However, this subject, important as it is, is just a subset of a larger subject called convex analysis. In this chapter, we shall give a brief introduction to this broader subject. In particular, we shall prove a few of the fundamental results of convex analysis and see that their proofs depend on some of the theory of linear programming that we have already developed.

Keywords

Feasible Solution Linear Programming Problem Convex Combination Convex Analysis Slack Variable 
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Notes

  1. Carathéodory, C. (1907), `Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen’, Mathematische Annalen 64, 95–115. 171Google Scholar
  2. Farkas, J. (1902), Theorie der einfachen Ungleichungen’, Journal für die reine und angewandte Mathematik 124, 1–27. 171Google Scholar
  3. Gordan, P. (1873), `Über die Auflösung linearer Gleichungen mit reelen Coefficienten’, Mathematische Annalen 6, 23–28. 171Google Scholar
  4. Stiemke, E. (1915), `Über positive Lösungen homogener linearer Gleichungen’, Mathematische Annalen 76, 340–342. 171Google Scholar
  5. Ville, J. (1938), Sur la théorie général des jeux ou intervient l’habileté des jouers, in E. Borel, ed., `Traité du Calcul des Probabilités et des ses Applications’, Paris, Gauthiers-Villars. 171Google Scholar
  6. Tucker, A. (1956), `Dual systems of homogeneous linear equations’, Annals of Mathematics Studies 38, 3–18. 171, 368Google Scholar
  7. Rockafellar, R. (1970), Convex Analysis, Princeton University Press, Princeton, NJ.Google Scholar

Copyright information

© Robert J. Vanderbei 2001

Authors and Affiliations

  • Robert J. Vanderbei
    • 1
  1. 1.Dept. of Operations Research & Financial EngineeringPrinceton UniversityUSA

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