Sharp Lipschitz Constants for Basic Optimal Solutions and Basic Feasible Solutions of Linear Programs

Abstract

The main purpose of this paper is to give Lipschitz constants for basic optimal solutions (or vertices of solution sets) and basic feasible solutions (or vertices of feasible sets) of linear programs with respect to right-hand side perturbations. The Lipschitz constants are given in terms of norms of pseudoinverses of submatrices of the matrices involved, and are sharp under very general assumptions. There are two mathematical principles involved in deriving the Lipschitz constants: (1) the local upper Lipschitz constant of a Hausdorff lower semicontinuous mapping is equal to the Lipschitz constant of the mapping and (2) the Lipschitz constant of a finite-set-valued mapping can be inherited by its continuous submappings. Moreover, it is proved that any Lipschitz constant for basic feasible solutions can be used as an Lipschitz constant for basic optimal solutions, feasible solutions, and optimal solutions.