Infinite Horizon Linear Programming With One-Dimensional Control Variable

We provide a set of sufficient conditions for the optimal value of an infinite horizon linear programming problem with one-dimensional control variable to be equal to the optimal value of its implied infinite horizon dual linear programming problem. The simplicity of this characterization is entirely due to the fact that in each period, there is only one inequality constraint that the control variable is required to satisfy. We show this, by introducing a generalized version of the model. In this more general framework the duality gap problem can be partially resolved via a limiting argument. However, if the solution values of the dual linear programming problems of the truncated “fixed end-point linear programming problems” solved by the optimal solution are bounded above, and a transversality condition is satisfied, then the duality gap problem can be resolved.