On the Stability of the Extreme Point Set in Linear Optimization
Miguel Angel Goberna, Mercedes Larriqueta, Virginia N. Vera de Serio · SIAM Journal on Optimization · 2005
This paper analyzes the stability properties of the set of extreme points of a closed convex set described by means of a given linear inequality system $\sigma$. We assume that all the coefficients of $\sigma$ can be arbitrarily perturbed maintaining the (possibly infinite) index set as well as the (finite) dimension of the space of variables, and we measure the size of these perturbations by means of the pseudometric of the uniform convergence. The paper characterizes the nonemptiness of the extreme point set under sufficiently small perturbations and the Berge lower semicontinuity of the extreme point set mapping at $\sigma$. It also gives necessary as well as sufficient conditions for the closedness and for the Berge upper semicontinuity at $\sigma$ which provide characterizations of these properties for finite systems.