Characterization of perturbed mathematical programs and interval analysis

Several authors have used interval arithmetic to deal with parametric or sensitivity analysis in mathematical programming problems. Several reported computational experiments have shown how interval arithmetic can provide such results. However, there has not been a characterization of the resulting solution interval in terms of the usual sensitivity analysis results. This paper presents a characterization of perturbed convex programs and the resulting solution intervals. Interval arithmetic was developed as a mechanism for dealing with the inherent error associated with numerical computations using a computational device. Here it is used to describe error in the parameters. The authors show that, for convex programs, the resulting solution intervals can be characterized in terms of the usual sensitivity analysis results. It has been often reported in the literature that even well behaved convex problems can exhibit pathological behavior in the presence of data perturbations. This paper uses interval arithmetic to deal with such problems, and to characterize the behavior of the perturbed problem in the resulting interval. These results form the foundation for future computational studies using interval arithmetic to do nonlinear parametric analysis.