# On the average number of steps of the simplex method of linear programming

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## Abstract

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 Words

Linear Programming Simplex Method Complexity Theory Algorithms Linear Complementarity Problem Path Following## Preview

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### References

- [1]I. Adler and S. Berenguer, “Random linear programs”, Technical Report ORC 81-4, Operations Research Center, University of California (Berkeley, 1981).Google Scholar
- [2]C. Allendoerfer and A. Weil, “The Gauss-Bonnet theorem for Riemannian polyhedra”, Transactions of the American Mathematical Society 53 (1943) 101–129.MATHCrossRefMathSciNetGoogle Scholar
- [3]K.-H. Borgwardt, “The average number of steps required by the simplex method is polynomial”, to appear in Zeitschrift für Operations Research.Google Scholar
- [4]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
- [5]G.B. Dantzig, Linear programming and extensions (Princeton University Press, Princeton, NJ, 1963).MATHGoogle Scholar
- [6]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
- [7]B.C. Eaves and H. Scarf, “The solution of systems of piecewise linear equations”, Mathematics of Operations Research 1 (1976) 1–27.MATHMathSciNetCrossRefGoogle Scholar
- [8]W. Feller, An introduction to probability theory and its applications, 2nd ed., Vol. 1 (Wiley, New York, 1957).MATHGoogle Scholar
- [9]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
- [10]M. Hirsch and S. Smale, “On algorithms for solving
*f(x)*=0”, Communications on Pure and Applied Mathematics 32 (1979) 281–312.MATHMathSciNetGoogle Scholar - [11]R. Howe, “Linear complementarity and the degree of mappings”, Cowles Foundation Discussion Paper No. 452, Yale University (New Haven, CT, 1980).Google Scholar
- [12]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
- [13]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
- [14]L. Lovász, “A new linear programming algorithm—Better or worse than the simplex method?”, The Mathematical Intelligencer 2 (1980) 141–146.MATHGoogle Scholar
- [15]J. May and R. Smith, “Random polytopes: Their definition, generation, and aggregate properties, Mathematical Programming 24 (1982) 39–54.MATHCrossRefMathSciNetGoogle Scholar
- [16]K.G. Murty, “Computational complexity of complementary pivot methods”, Mathematical Programming Study 7 (1978) 61–73.MATHMathSciNetGoogle Scholar
- [17]M. Shub and S. Smale, “Computational complexity: On the geometry of polynomials and a theory of cost, Part I”, preprint (1982).Google Scholar
- [18]S. Smale, “A convergent process of price adjustment and global Newton methods”, Journal of Mathematical Economics 3 (1976) 107–120.MATHCrossRefMathSciNetGoogle Scholar
- [19]S. Smale, “The fundamental theorem of algebra and complexity theory”, Bulletin of the American Mathematical Society 4 (1981) 1–36.MATHMathSciNetCrossRefGoogle Scholar
- [20]J. Traub and H. Woźniakowski, “Complexity of linear programming”, Operations Research Letters 1 (1982) 59–62.MATHCrossRefGoogle Scholar
- [21]P. Wolfe, “The ellipsoid algorithm” (Letter to the Editor), Science 208 (1980) 240–242.MathSciNetGoogle Scholar

## Copyright information

© The Mathematical Programming Society, Inc. 1983