Structure of Solutions for Continuous Linear Programs with Constant Coefficients

Abstract

We consider continuous linear programs over a continuous finite time horizon $T$, with linear cost coefficient functions, linear right-hand side functions, and a constant coefficient matrix, as well as their symmetric dual. We search for optimal solutions in the space of measures or of functions of bounded variation. These models generalize the separated continuous linear programming models and their various duals, as formulated in the past by Anderson, by Pullan, and by Weiss. In a recent paper, we have shown that under a Slater-type condition, these problems possess optimal strongly dual solutions. In this paper, we give a detailed description of optimal solutions and define a combinatorial analogue to basic solutions of standard LP. We also show that feasibility implies existence of strongly dual optimal solutions without requiring the Slater condition. We present several examples to illustrate the richness and complexity of these solutions.