Constrained Entropy Models; Solvability and Sensitivity

Abstract

The paper presents an analysis of the constrained entropy maximization model from the point of view of geometric programming. While the original entropy maximization model consists of maximizing the entropy of a system subject only to constraints that the solution be a probability measure, the models considered here contain an additional set of linear constraints. These constrained models have been the subject of a wide range of applications in transportation and geographical analysis. Using the duality theory of geometric programming, we develop the dual to the constrained model, which as in the case of the original model is unconstrained except for the positivity restrictions on the dual variables. In addition, this duality theory enables us to study the solvability of the model and the impact of changes in the model parameters on the solution. The sensitivity analysis provides approximations to the optimal solution to problems with perturbed data without requiring the re-solving of the model. This analysis is appropriate for changes in the right hand sides of the constraints, the coefficients in the constraints, and the objective function coefficients. Since the constraint coefficients correspond to the objective function exponents in the primal geometric program, the analysis provides a means of studying such changes is any geometric program. The computational aspects of the procedures are illustrated on a trip distribution problem.