On the Stability of the Boundary of the Feasible Set in Linear Optimization

This paper analizes the relationship between the stability properties of the closed convex sets in finite dimensions and the stability properties of their corresponding boundaries. We consider a given closed convex set represented by a certain linear inequality system σ whose coefficients can be arbitrarily perturbed, and we measure the size of these perturbations by means of the pseudometric of the uniform convergence. It is shown that the feasible set mapping is Berge lower semicontinuous at σ if and only if the boundary mapping satisfies the same property. Moreover, if the boundary mapping is semicontinuous in any sense (lower or upper; Berge or Hausdorff) at σ, then it is also closed at σ. All the mentioned stability properties are equivalent when the feasible set is a convex body.