Convex Analysis

  • Robert J. Vanderbei
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 114)


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.


Feasible Solution Linear Programming Problem Convex Combination Convex Analysis Slack Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


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

Copyright information

© Robert J.Vanderbei 2008

Authors and Affiliations

  • Robert J. Vanderbei
    • 1
  1. 1.Dept. of Operations Research and Financial EngineeringPrinceton UniversityNew JerseyUSA

Personalised recommendations