Asymptotic Linear Programming

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 t₀ 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 t₀ (the first functional form valid for t < t₀, the second for t < t₀)]. 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.