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On Duality Theory of Conic Linear Problems

  • Alexander Shapiro
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 57)

Abstract

In this paper we discuss duality theory of optimization problems with a linear objective function and subject to linear constraints with cone inclusions, referred to as conic linear problems. We formulate the Lagrangian dual of a conic linear problem and survey some results based on the conjugate duality approach, where the questions of”no duality gap” and existence of optimal solutions are related to properties of the corresponding optimal value function. We discuss in detail applications of the abstract duality theory to the problem of moments, linear semi-infinite, and continuous linear programming problems.

Keywords

Banach Space Dual Problem Duality Theory Topological Vector Space Primal Problem 
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.

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References

  1. [1]
    E. J. Anderson and P. Nash. Linear Programming in Infinite-Dimensional Spaces, Theory and Applications, Wiley, 1987.MATHGoogle Scholar
  2. [2]
    R.E. Bellman. Bottleneck problems and dynamic programming, Proceed ings of the National Academic of Sciences of the USA, 39: 947–951, 1953.MATHCrossRefGoogle Scholar
  3. [3]
    P. Billingsley. Convergence of Probability Measures (2nd ed.), Wiley, 1999.MATHCrossRefGoogle Scholar
  4. [4]
    J.F. Bonnans and A. Shapiro. Perturbation Analysis of Optimization Problems, Springer-Verlag, 2000.MATHGoogle Scholar
  5. [5]
    I. Ekeland and R. Temam. Convex Analysis and Variational Problems, North-Holland, 1976.MATHGoogle Scholar
  6. [6]
    K. Glashoff and S.A. Gustafson. Linear Optimization and Approximation, Springer-Verlag, 1983.MATHCrossRefGoogle Scholar
  7. [7]
    M. A. Goberna and M. A. Lopez. Linear Semi-Infinite Optimization, Wiley, 1998.MATHGoogle Scholar
  8. [8]
    R.C. Grinold. Symmetric duality for continuous linear programs, SIAM Journal on Applied Mathematics, 18:84–97, 1970.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    R. Hettich and K.O. Kortanek. Semi-infinite programming: theory, meth ods and applications, SIAM Review, 35: 380–429, 1993.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    A. Hoffman. On approximate solutions of systems of linear inequalities, Journal of Research of the National Bureau of Standards, Section B, Math ematical Sciences, 49: 263–265, 1952.Google Scholar
  11. [11]
    R.B. Holmes. Geometric Functional Analysis and its Applications, Springer-Verlag, 1975.MATHCrossRefGoogle Scholar
  12. [12]
    K. Isii. On sharpness of Tchebycheff type inequalities, Annals of the In state of Statistical Mathematics, 14: 185–197, 1963.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    I.V. Kantorovich and G.P. Akilov. Functional Analysis, Nauka, Moscow, 1984.MATHGoogle Scholar
  14. [14]
    J.H.B. Kemperman. The general moment problem, a geometric approach, Annals of Matliematics and Statistics, 39: 93–122, 1968.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    N. Levinson. A class of continuous linear programming problems, Journal of Mathematical Analysis and Applications, 16: 78–83, 1966.MathSciNetCrossRefGoogle Scholar
  16. [16]
    HJ. Landau (editor), Moments in mathematics, Proc. Sympos. Appl. Math., 37, Amer. Math. Soc., Providence, RI, 1987.MATHGoogle Scholar
  17. [17]
    M. C. Pullan. A duality theory for separated continuous linear programs, SIAM Journal on Control and Optimization, 34: 931–965, 1996.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    M. C. Pullan. Existence and duality theory for separated continuous linear programs, Mathematical Models and Systems, 3: 219–245, 1997.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    S.M. Robinson. Regularity and stability for convex multivalued functions, Mathematics of Operations Research, 1: 130–143, 1976.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    R.T. Rockafellar. Convex Analysis, Princeton University Press, 1970.MATHGoogle Scholar
  21. [21]
    R.T. Rockafellar. Integrals which are convex functionals. II, Pacific Jour nal of Mathematics, 39: 439–469, 1971.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    R.T. Rockafellar. Conjugate Duality and Optimization, Volume 16 of Regional Conference Series in Applied Mathematics, SLAM, 1974.MATHCrossRefGoogle Scholar
  23. [23]
    W.W. Rogosinsky. Moments of non-negative mass, Proceedings of the Royal Society London, Serie A, 245: 1–27, 1958.CrossRefGoogle Scholar
  24. [24]
    J.E. Smith. Generalized Chebychev inequalities: theory and applications in decision analysis, Operations Research, 43: 807–825, 1995.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Alexander Shapiro
    • 1
  1. 1.School of Industrial and Systems EngineeringGeorgia Institute of TechnologyUSA

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