Stability of the extreme point set of a polyhedron

This paper is focused on the stability properties of the extreme point set of a polyhedron. We consider a polyhedral setX(A,b) which is defined by a linear system of equality and inequality constraintsAxb, where the matrixA and the right-hand sideb are subject to perturbations. The extreme point setE(X(A,b)) of the polyhedronX(A,b) defines a multivalued map :(A,b)E(X(A,b)). In the paper, characterization of continuity and Lipschitz continuity of the map is obtained. Boundedness of the setX(A,b) is not assumed It is shown that lower Lipschitz continuity is equivalent to the lower semicontinuity of the map and to the Robinson and Mangasarian-Fromovitz constraint qualifications. Upper Lipschitz continuity is proved to be equivalent to the upper semicontinuity of the map . It appears that the upper semicontinuity of the map implies the lower semicontinuity of this map. Some examples of using the conditions obtained are provided.The author wishes to thank Dr. N. M. Novikova, Dr. S. K. Zavriev, and anonymous referees for their helpful comments and advice. The research described in this publication was made possible in part by Grant NJCU100 from the International Science Foundation and Russian Government, and by the Euler Grant, Deutsche Mathematiker Vereinigung.