Mathematical Methods of Operations Research

, Volume 64, Issue 3, pp 541–555 | Cite as

Asymptotic sign-solvability, multiple objective linear programming, and the nonsubstitution theorem

  • L. Cayton
  • R. Herring
  • A. HolderEmail author
  • J. Holzer
  • C. Nightingale
  • T. Stohs
Original Article


In this paper we investigate the asymptotic stability of dynamic, multiple-objective linear programs. In particular, we show that a generalization of the optimal partition stabilizes for a large class of data functions. This result is based on a new theorem about asymptotic sign-solvable systems. The stability properties of the generalized optimal partition are used to address a dynamic version of the nonsubstitution theorem.


Multiple-objective linear programming Asymptotic programming Sign-solvability Nonsubstitution theorem Computational economics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Altman E, Avrachenkov K, Filar J (1999) Asymptotic linear programming and policy improvement for singularly perturbed markov decision processes. Math Methods Oper Res 59(1):97–109MathSciNetGoogle Scholar
  2. Bernard L (1989) A generalized inverse method for asymptotic linear programming. Math Program 43(1):71–86MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bernard L (1993) An efficient basis update for asymptotic linear programming. Linear Algebra Appl 184:83–102MathSciNetCrossRefzbMATHGoogle Scholar
  4. Brualdi R, Shader B (1995) Matrices and sign-solvable linear systems. Cambridge University Press, New YorkGoogle Scholar
  5. Campbell S, Meyer C Jr (1979) Generalized inverses of linear transformations. Fearon Pitman Publishers Inc., BelmontzbMATHGoogle Scholar
  6. Caron R, Greenberg H, Holder A (2002) Analytic centers and repelling inequalities. Eur J Oper Res 143(2):268–290zbMATHMathSciNetCrossRefGoogle Scholar
  7. Chander P (1974) A simple proof of the nonsubstitution theorem. Q J Econ 88:698–701zbMATHCrossRefGoogle Scholar
  8. Diewart W (1975) The samuelson nonsubstitution theorem and the computation of equilibrium prices. Econometrica 43(1):57–64MathSciNetCrossRefGoogle Scholar
  9. Ehrgott M (2000) Multicriteria optimization. In: Lecture notes in economics and mathematical systems, vol. 491. Springer, Berlin Heidelberg New YorkGoogle Scholar
  10. Fujimoto T (1987) A simple proof of the nonsubstitution theorem. J Quant Econ 3(1):35–38Google Scholar
  11. Greenberg H (1996–2001) Mathematical programming glossary. Scholar
  12. Grunbaum B (1967) Convex polytopes. Wiley-Interscience, New YorkGoogle Scholar
  13. Hasfura-Buenaga J, Holder A, Stuart J (2005) The asymptotic optimal partition and extensions of the nonsubstitution theorem. Linear Algebra Appl 394:145–167zbMATHMathSciNetCrossRefGoogle Scholar
  14. Holder A (2001) Partitioning multiple objective solutions with applications in radiotherapy design. Technical report~54, Trinity University Mathematics. Optim Eng (to appear)Google Scholar
  15. Jeroslow R (1972) Asymptotic linear programming. Oper Res 21:1128–1141MathSciNetCrossRefGoogle Scholar
  16. Jeroslow R (1973) Linear programs dependent on a single parameter. Discrete Math 6:119–140zbMATHMathSciNetCrossRefGoogle Scholar
  17. Kuga K (2001) The non-substitution theorem: multiple primary factors and the cost function approach. Technical report discussion paper no. 529, The Institute of Social and Economic Research, Osaka Univeristy, Osaka, JapanGoogle Scholar
  18. Kurz H, Salvadori N (1995) Theory of production: a long-period analysis. Cambridge University Press, New YorkGoogle Scholar
  19. Mirrlees J (1969) The dynamic nonsubstitution theorem. Rev Econ Stud 36(105):67–76Google Scholar
  20. Roos C, Terlaky T, Vial J-Ph (1997) Theory and algorithms for linear optimization: an interior point approach. Wiley, New YorkzbMATHGoogle Scholar
  21. Samuelson P (1951) Abstract of a theorem concerning substitutability in open leontief models. In: Koopmans T (eds). Activity analysis of production and allocation, Chap 7. Wiley, New YorkGoogle Scholar
  22. Smith A (1991) The wealth of nations. Alfred A. Knopf, Inc., New YorkGoogle Scholar
  23. Ying H (1996) A canonical form for pencils of matrices with applications to asymptotic linear programs. Linear Algebra Appl 234:97–123MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • L. Cayton
    • 1
  • R. Herring
    • 2
  • A. Holder
    • 3
    Email author
  • J. Holzer
    • 4
  • C. Nightingale
    • 5
  • T. Stohs
    • 6
  1. 1.University of CaliforniaSan Diego, La JollaUSA
  2. 2.Oberlin CollegeOberlinUSA
  3. 3.Trinity University MathematicsSan AntonioUSA
  4. 4.University of WisconsinMadisonUSA
  5. 5.Mills CollegeOaklandUSA
  6. 6.University of Nebraska-LincolnLincolnUSA

Personalised recommendations