Abstract
A decomposition algorithm using Lemke's method is proposed for the solution of quadratic programming problems having possibly unbounded feasible regions. The feasible region for each master program is a generalized simplex of minimal size. This property is maintained by a dropping procedure which does not affect the finiteness of the convergence. The details of the matrix transformations associated with an efficient implementation of the algorithm are given. Encouraging preliminary computational experience is presented.
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References
P. Broise, P. Huard and J. Sentenac,Décomposition des programmes mathématiques (Dunod, Paris, 1968).
D.G. Cantor and M. Gerla, “Optimal routing in a packet-switched computer network”,IEEE Transaction on Computers C-23 (1974) 1062–1069.
R.W. Cottle and G.B. Dantzig, “Complementary pivot theory of mathematical programming”,Linear Algebra and its Applications 1 (1968) 103–125.
G.B. Dantzig,Linear programming and extensions (Princeton University Press, Princeton, NJ, 1963).
G.B. Dantzig and P. Wolfe, “The decomposition algorithm for linear programming”,Operations Research 8 (1960) 101–111.
A.M. Geoffrion, “Elements of large-scale mathematical programming: parts I and II”,Management Science 16 (1970) 652–691.
R. Hartman and P. Reiser, “Über den Aufwand zür lösung quadratischer Optimierungsprobleme mit den Algorithmen von Beale und von Lemke”,Zeitschrift für Operations Research 18 (1974) B57-B63.
B. von Hohenbalken, “A finite algorithm to maximize certain pseudoconcave functions on polytopes”,Mathematical Programming 8 (1975) 189–206.
B. von Hohenbalken, “Simplicial decomposition in nonlinear programming algorithms”,Mathematical Programming 13 (1977) 49–68.
C.A. Holloway, “An extension of the Frank and Wolfe method of feasible directions”,Mathematical Programming 6 (1974) 14–27.
L.S. Lasdon,Optimization theory for large systems (Macmillan Co., New York, 1970).
L.G. Proll, “Remark on algorithm 431 [H]: A computer routine for quadratic and linear programming problems”,Communications of the Association for Computing Machinery 17 (1974) 590.
A. Ravindrin, “Algorithm 431: A computer routine for quadratic and linear programming problems”,Communications of the Association for Computing Machinery 15 (1972) 818–820.
R.T. Rockafellar,Convex analysis (Princeton University Press, Princeton, NJ, 1970).
J.A. Tomlin, “Robust implementation of Lemke's method for the linear complementarity problem”,Mathematical Programming Study 7 (1978) 55–60.
P. Wolfe, “Methods of nonlinear programming”, in: J. Abadie, ed.,Nonlinear programming (Wiley, New York, 1967).
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Sacher, R.S. A decomposition algorithm for quadratic programming. Mathematical Programming 18, 16–30 (1980). https://doi.org/10.1007/BF01588293
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DOI: https://doi.org/10.1007/BF01588293