Asymptotic Linear Programming

Robert G. Jeroslow
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Robert G. Jeroslow: Carnegie-Mellon University, Pittsburgh, Pennsylvania

Operations Research, 1973, vol. 21, issue 5, 1128-1141

Abstract: This paper studies the linear programming problem in which all coefficients (even those of the stipulations matrix) are rational functions of a single parameter t called “time,” and provides an algorithm that can solve problems of the following two types: (1) Steady-state behavior [the algorithm can be used to determine the functional form x ( t ) of the optimal solution as a function of t , this form being valid for all “sufficiently large” values of t ], and (2) sensitivity analysis [if a value t 0 of “time” is given, the algorithm can be used to determine the two possible functional forms of the optimal solution for all values of t “sufficiently close” to t 0 (the first functional form valid for t t 0 , the second for t t 0 )]. In addition, the paper gives certain qualitative information regarding steady-state behavior, including the following result: If for some one of the properties of consistency, boundedness, or bounded constraint set, there exists a sequence t n ↗ +∞ such that the linear program at n has this property for all n , then the program has this property for all “sufficiently large” values of t .

Date: 1973
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