Abstract
We extend and simplify Smale's work on the expected number of pivots for a linear program with many variables and few constraints. Our analysis applies to new versions of the simplex algorithm and to new random distributions.
Similar content being viewed by others
References
I. Adler and S. Berenguer, “Random linear programs”, Operations Research Center Report 81-4, University of California (Berkeley, CA, 1981).
I. Adler, R. Karp and R. Shamir, “A simplex variant solving anm-by-d linear program in O(min(m2, d2)) expected number of steps”, Report UCB CSD 83/157, Computer Science Division, University of California (Berkeley, CA, December 1983).
J. Bentley, H. Kung, M. Schkolnick and C. Thompson, “On the average number of maxima in a set of vectors“,Journal of the ACM 25 (1978) 536–543.
K.-H. Borgwardt, “Some distribution-independent results about the asymptotic order of the average number of pivot steps of the simplex-method“,Mathematics of Operations Research 7 (1982) 441–462.
M. Haimovich, “The simplex algorithm is very good!” Business Department Technical Report, Columbia University (New York, NY, April 1983).
H. Kung, F. Luccio and F. Preparata, “On finding maxima of a set of vectors“,Journal of the ACM 22 (1975) 469–476.
J. May and R. Smith, “Random polytopes: Their definitions, generation and aggregate properties“,Mathematical Programming 24 (1982) 39–54.
N. Megiddo, “Solving linear-programming linear-time when the dimension is fixed“,Journal of the ACM 31 (1984) 114–127.
B. O'Neill, “The number of outcomes in the Pareto-optimal set of discrete bargaining games“,Mathematics of Operations Research 6 (1981) 571–578.
S. Smale, “On the average number of steps of the simplex method of linear programming“,Mathematical Programming 27 (1983) 241–263.
S. Smale, “The problem of the average speed of the simplex method“, in: A. Bachem, M. Grötschel and B. Korte, eds.,Mathematical programming: The state of the art (Springer-Verlag, Berlin and New York, 1983), pp. 530–539.
F. Yao, “On finding maximal elements in a set of plane vectors”, Computer Science Technical Report R-74-667, University of Illinois (Urbana, IL, 1974).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Blair, C. Random linear programs with many variables and few constraints. Mathematical Programming 34, 62–71 (1986). https://doi.org/10.1007/BF01582163
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01582163