Calmness modulus of fully perturbed linear programs

This paper provides operative point-based formulas (only involving the nominal data, andnot data in a neighborhood) for computingor estimating the calmness modulus of the optimal set (argmin) mapping in linear optimization under uniqueness of nominal optimal solutions. Our analysis is developed in two different parametric settings. First, in the framework of canonical perturbations (i.e., perturbations of the objective function and the right-hand-side of the constraints), the paper provides a computationally tractable formula for the calmness modulus, which goes beyond some preliminary results of the literature. Second, in the framework of full perturbations (perturbations of all coefficients), after characterizing the calmness property for the optimal set mapping, the paper provides an operative upper bound for the corresponding calmness modulus, as well as some illustrative examples. We provide This research has been partially supported by Grants MTM2011-29064-C03-03 and MTM2014-59179C2-2-P from MINECO, Spain. The research of the second author is also partially supported by Fondecyt Project No. 1151003, ECOS-Conicyt Project No. C10E08, and Math-Amsud No. 13MATH-01 2013. B J. Parra [email protected] M. J. Cánovas [email protected] A. Hantoute [email protected] F. J. Toledo [email protected] 1 Center of Operations Research, Miguel Hernández University of Elche, 03202 Elche, Alicante, Spain 2 Departamento de Ingeniería Matemática, Centro de Modelamiento Matemático (CMM), Universidad de Chile, Santiago, Chile