ISSN:
1573-2878
Keywords:
Polyhedral sets
;
extreme points
;
multivalued maps
;
continuity
;
stability
;
linear programming
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract 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 constraintsAx≤b, 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 ℳ:(A,b)→E(X(A,b)). In the paper, characterization of continuity and Lipschitz continuity of the map ℳ 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 ℳ and to the Robinson and Mangasarian-Fromovitz constraint qualifications. Upper Lipschitz continuity is proved to be equivalent to the upper semicontinuity of the map ℳ. It appears that the upper semicontinuity of the map ℳ implies the lower semicontinuity of this map. Some examples of using the conditions obtained are provided.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02190003
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