A closed formula for local sensitivity analysis in mathematical programming

Abstract

This article introduces a method for local sensitivity analysis of practical interest. A theorem is given that provides a general and neat manner to obtain all sensitivities of a general nonlinear programming problem (around a local minimum) with respect to any parameter irrespective of it being a right-hand side, objective function or constraint constant. The method is based on the well-known duality property of mathematical programming, which states that the partial derivatives of the primal objective function with respect to the constraints' right-hand side parameters are the optimal values of the dual problem variables. For the parameters or data for which sensitivities are sought to appear on the right-hand side, they are converted into artificial variables and set to their actual values, thus obtaining the desired constraints. If the problem is degenerated and partial derivatives do not exist, the method also permits obtaining the right, left, and also directional derivatives, if they exist. In addition to its general applicability, the method is also computationally inexpensive because the necessary information becomes available without extra calculations. Moreover, analytical relations among sensitivities, locally valid, are straightforwardly obtained. It is also shown how the roles of the objective function and any of the active constraints (equality or inequality) can be exchanged leading to equivalent optimization problems. This permits obtaining the sensitivities of any constraint with respect to the parameters without the need of repeating the calculations. The method is illustrated by its application to two examples, one degenerated and the other one of a competitive market.

Keywords:

• Local sensitivity
• Mathematical programming
• Duality

Acknowledgements

The authors are indebted to the Ministry of Science and Technology of Spain (Projects DPI2002-04172-C04-02 and DPI2003-01362), the Ministry of Education of Spain and the Fulbright Commission for partial support.