Infinite Horizon Linear Programming With One-Dimensional State Variable

We provide a set of sufficient conditions for the optimal value of an infinite horizon linear programming problem with one-dimensional state variable to be equal to the optimal value of its implied infinite horizon dual linear programming problem. Our sufficient conditions require that in every period the linear inequality determining the constraint for the state variable for the next period is expressed in terms of a “non-constant” function of the current value of the state variable, the state variable along the optimal trajectory is always strictly positive beginning with time period one and is strictly less than its upper-bound in “at least one” time period. In addition, the transversality condition we invoke is that the product in each period, of the state variable and the co-state or dual variable for the inequality constraining the state variable in that period, converges to zero. The simplicity of this characterization is entirely due to the fact that in each period, there is only one inequality constraint that the state variable is required to satisfy. We show this, by introducing a generalized infinite horizon linear programming problem with one-dimensional state variable. In this more general framework the duality gap problem can be partially resolved via a limiting argument.