Abstract
An algorithm for determining all the extreme points of a convex polytope associated with a set of linear constraints, via the computation of basic feasible solutions to the constraints, is presented. The algorithm is based on the product-form revised simplex method and as such can be readily linked onto standard linear programming codes. Applications of such an algorithm are reviewed and limited computational experience given.
Similar content being viewed by others
References
M.L. Balinski, “An algorithm for finding all vertices of convex polyhedral sets”,Journal of the Society Industrial and Applied Mathematics (1961) 72–88.
C. Berge,The theory of graphs and its applications (Methuen, London, 1962).
C.A. Burdet, “Generating all the faces of a polyhedron”,SIAM Journal on Applied Mathematics, 26 (1974) 479–489.
V.A. Cabot and R.L. Francis, “Solving certain nonconvex quadratic minimization problems by ranking the extreme points”,Operations Research, 18 (1970) 82–86.
A. Charnes, W.W. Cooper and A. Henderson,An introduction to linear programming (Wiley; New York, 1953).
N.V. Chernikova, “An algorithm for finding a general formula for non-negative solutions of a system of linear inequalities”,U.S.S.R. Computational Mathematics and Mathematical Physics 5 (1965) 228–233.
H. Greenberg, “An algorithm for determining redundant inequalities and all solutions to polyhedra”,Numerische Mathematik, 24 (1975) 19–26.
B. Grunbaum,Convex polytopes (Wiley, New York, 1967).
G. Hadley,Linear programming (Addison-Wesley, Reading, MA, 1962).
M.J.L. Kirby, H.L. Love and Kanti Swarup, “Extreme point mathematical programming”,Management Science, 18 (1972) 540–549.
D.E. Knuth,The art of computer programming, Vol. 2:Seminumerical algorithms (Addison-Wesley, Reading, MA, 1968).
M. Manas and J. Nedoma, “Finding all vertices of a convex polyhedron”,Numerische Mathematik, 12 (1968) 226–229.
T.H. Mattheis, “An algorithm for determining irrelevant constraints and all vertices in systems of linear inequalities”,Operations Research 21 (1973) 247–260.
T.S. Motzkin, H. Raiffa, G.L. Thompson and R.M. Thrall, “The double description method”, in: H.W. Kuhn and A.W. Tucker, eds.,Contributions to the theory of games, Vol. 2, (Princeton University Press, Princeton, RI, 1973).
K.G. Murty, “Solving the fixed charge problem by ranking the extreme points”,Operations Research 16 (1968) 268–279.
K.G. Murty, “An algorithm for ranking all the assignments in increasing order of costs”,Operations Research 16 (1969) 682–687.
M.A. Pollatschek and B. Avi-Itshak, “Sorting feasible basic solutions of a linear program”, presented at the 3rd Annual Israel Conference on Operations Research (1969).
D.S. Rubin, “Vertex generation and cardinality constrained linear programs”,Operations Research 23 (1975) 555–564.
G.J. Silverman, “Computational considerations in extreme point enumeration”. IBM Los Angeles Scientific Center, Report G320-2649 (1971).
W.J. Sullivan and E. Koenigsberg, “Mixed integer programming applied to ship allocation”, in: E.M.L. Beale, ed.,Applications of mathematical programming techniques, (English Universities' Press, London, 1970).
H. Uzawa, “A theorem on convex polyhedral cones”, in: Arrow, Hurwicz and Uzawa, eds.,Studies in linear and nonlinear programming, (Standford University Press, Stanford, CA, 1958).
S. Vajda, “Manpower planning and mathematical programming”, paper presented at the Operational Research Society annual conference, Brighton (1974).
C. Van De Panne, “A node method for multiparametric linear programming”,Management Science 21 (1975) 1014–1020.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dyer, M.E., Proll, L.G. An algorithm for determining all extreme points of a convex polytope. Mathematical Programming 12, 81–96 (1977). https://doi.org/10.1007/BF01593771
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01593771