On the average number of steps of the simplex method of linear programming
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The goal is to give some theoretical explanation for the efficiency of the simplex method of George Dantzig. Fixing the number of constraints and using Dantzig's self-dual parametric algorithm, we show that the number of pivots required to solve a linear programming problem grows in proportion to the number of variables on the average.
Key WordsLinear Programming Simplex Method Complexity Theory Algorithms Linear Complementarity Problem Path Following
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- I. Adler and S. Berenguer, “Random linear programs”, Technical Report ORC 81-4, Operations Research Center, University of California (Berkeley, 1981).Google Scholar
- K.-H. Borgwardt, “The average number of steps required by the simplex method is polynomial”, to appear in Zeitschrift für Operations Research.Google Scholar
- R.W. Cottle and G.B. Dantzig, “Complementary pivot theory of mathematical programming”, in: G.B. Dantzig and A.F. Veinott, Jr., eds., Mathematics of the Decision Sciences, Part 1 (American Mathematical Society, Providence, RI, 1968) pp. 115–136.Google Scholar
- G.B. Dantzig, “Expected number of steps of the simplex method for a linear program with a convexity constraint”, Technical Report SOL 80-3, Systems Optimization Laboratory, Department of Operations Research, Stanford University (Stanford, CA, 1980).Google Scholar
- D. Gale, H.W. Kuhn, and A.W. Tucker, “Linear programming and the theory of games”, in: T.C. Koopmans, ed., Activity analysis of production and allocation (Wiley, New York, 1951).Google Scholar
- R. Howe, “Linear complementarity and the degree of mappings”, Cowles Foundation Discussion Paper No. 452, Yale University (New Haven, CT, 1980).Google Scholar
- V. Klee and G. Minty, “How good is the simplex algorithm?”, in: O. Shisha, ed., Inequalities III (Academic Press, New York, 1972) pp. 159–175.Google Scholar
- T.M. Liebling, “On the number of iterations of the simplex method”, in: R. Henn, H. Künzi, and H. Schubert, eds., Operations Research Verfahren/Methods of Operations Research XVII (1973) 248–264.Google Scholar
- M. Shub and S. Smale, “Computational complexity: On the geometry of polynomials and a theory of cost, Part I”, preprint (1982).Google Scholar