Abstract
This note shows that, if a parametric linear program, min{cx: Ax = b 1 θ+b 2,x≥0}, has optimal solutions in an interval\(\left[ {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\theta } ,\bar \theta } \right]\) forθ, then, depending on degeneracy, the solutions(θ) is a continuous vector function or a continuous point-to-set mapping. In the latter case an algorithm is introduced to solve the problem and generate a continuous vector solution in\(\left[ {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\theta } ,\bar \theta } \right]\).
Similar content being viewed by others
References
M.S. Bazaraa and C.M. Shetty,Nonlinear Programming: Theory and Algorithms (Wiley, New York, 1979).
R.G. Bland, “New finite pivoting rules for the Simplex Method,”Mathematics of Operations Research 2 (1977) 103–107.
W. Orchard-Hays,Advanced Linear Programming Computing Techniques (McGraw-Hill, New York, 1968).
M. Simmonard,Linear Programming (Prentice-Hall, Englewood Cliffs, NJ, 1966).
X.-S. Zhang, “Solution property of parametric linear programs,” Tech. Report 83-7, Computer Science Department, University of Minnesota (Minneapolis, MN, 1983).
Author information
Authors and Affiliations
Additional information
Part of this author's work was done while he visited the Computer Science Department, University of Minnesota, Minneapolis, and was supported by the National Science Foundation under the research grant MCN 81-01214.
Rights and permissions
About this article
Cite this article
Zhang, XS., Liu, DG. A note on the continuity of solutions of parametric linear programs. Mathematical Programming 47, 143–153 (1990). https://doi.org/10.1007/BF01580857
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01580857