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Improved asymptotic analysis of the average number of steps performed by the self-dual simplex algorithm.(English)Zbl 0618.90061

Math. Program.35, 140-172 (1986).

We analyze the average number of steps performed by the self-dual simplex algorithm for linear programming, under the probabilistic model of spherical symmetry. The model was proposed byS. Smale [ibid. 27, 241-262 (1983;Zbl 0526.90060)]. Consider a problem of n variables with m constraints. Smale established that for every number of constraints m, there is a constant c(m) such that the number of pivot steps of the self- dual algorithm, \(\rho\) (m,n), is less than c(m)(ln n)\({}^{m(m+1)}\). We improve upon this estimate by showing that \(\rho\) (m,n) is bounded by a function of m only. The symmetry of the function in m and n implies that \(\rho\) (m,n) is in fact bounded by a function of the smaller of m and n.

Cited in1 Review

Cited in12 Documents

MSC:

90C05	Linear programming
68Q25	Analysis of algorithms and problem complexity

Keywords:

probabilistic analysis;average number of steps;self-dual simplex algorithm;spherical symmetry

Citations:

Zbl 0526.90060

Cite Review PDF

Full Text:DOI

References:

[1]	I. Adler, ”The expected number of pivots needed to solve parametric linear programs and the efficiency of the Self-Dual Simplex method”, Technical Report, Department of Industrial Engineering and Operations Research, University of California, (Berkeley, CA, June 1983).
[2]	I. Adler, R.M. Karp and R. Shamir, ”A family of simplex variants solving anm {\(\times\)} d linear program in expected number of pivot steps depending ond only”, Report UCB CSD 83/157, Computer Science Division, University of California, (Berkeley, CA, December 1983). ·Zbl 0618.90064
[3]	I. Adler, R.M. Karp and R. Shamir, ”A simplex variant solving anm {\(\times\)} d linear program in O(min(m 2,d 2)) expected number of steps”, Report UCB CSD 83/158, Computer Science Division, University of California (Berkeley, CA, December 1983). ·Zbl 0641.65054
[4]	I. Adler and N. Megiddo, ”A simplex-type algorithm solves linear programs of orderm {\(\times\)} n in only O((min(m, n))2) steps on the average”, preliminary report, November 1983.
[5]	I. Adler and N. Megiddo, ”A simplex algorithm whose average number of steps is bounded between two quadratic functions of the smaller dimension”, in:Proceedings of the 16th Annual ACM Symposium on Theory of Computing (ACM, New York, NY, May 1984) pp. 312–323. Also,Journal of the Association for Computing Machinery 32 (1985) (to appear).
[6]	I. Adler, N. Megiddo and M.J. Todd, ”New results on the average behavior of simplex algorithms”,Bulletin of the American Mathematical Society 11 (1984) 378–382. ·Zbl 0545.90066 ·doi:10.1090/S0273-0979-1984-15317-5
[7]	C. Blair, ”Random linear programs with many variables and few constraints”, Faculty Working Paper No. 946, College of Commerce and Business Administration, University of Illinois at Urbana-Champaign, (Urbana, IL, April 1983).
[8]	R.G. Bland, ”New finite pivoting rules”,Mathematics of Operations Research 3 (1978) 103–107. ·Zbl 0408.90050
[9]	K.-H. Borgwardt, ”Untersuchungen zur asymptotik der mittleren schriftzahl von simplexverfahren in der linearen optimierung”, Dissertation, Universität Kaiserlautern (1977). ·Zbl 0416.90041
[10]	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 Reserach 7 (1982) 441–462. ·Zbl 0498.90054 ·doi:10.1287/moor.7.3.441
[11]	K.-H. Borgwardt, ”The average number of steps required by the simplex method is polynomial”,Zeitschrift für Operations Research 26 (1982) 157–177. ·Zbl 0488.90047 ·doi:10.1007/BF01917108
[12]	G.B. Dantzig,Linear programming and extensions (Princeton University Press, Princeton, New Jersey, 1963).
[13]	M. Haimovich, ”The simplex algorithm is very good! - On the expected number of pivot steps and related properties of random linear programs”, Technical Report, Columbia University, (New York, NY, April 1983).
[14]	G. Kolata, ”Mathematician solves simplex problem”,Science 217 (1982) 39. ·doi:10.1126/science.217.4554.39
[15]	C.E. Lemke, ”Bimatrix equilibrium points and mathematical programming”,Management Science 11 (1965) 681–689. ·Zbl 0139.13103 ·doi:10.1287/mnsc.11.7.681
[16]	N. Megiddo, ”Linear programming in linear time when the dimension is fixed”,Journal of the Association for Computing Machinery 31 (1984) 114–127. ·Zbl 0637.90064
[17]	N. Megiddo, ”On the expected number of linar complementarity cones intersected by random and semi-random rays”,Mathematical Programming 35 (1986) 225–235. ·Zbl 0613.90092 ·doi:10.1007/BF01580648
[18]	N. Megiddo, ”A note on the generality of the self-dual simplex algorithm with various starting points”, in:Methods of operations research (Oelgeschlager, Gunn & Hain, 1985), to appear. ·Zbl 0561.90063
[19]	L. Santalo,Integral geometry and geometric probability (Addison-Wesley, Reading, MA, 1976).
[20]	S. Smale, ”On the average number of steps of the simplex method of linear programming”,Mathematical Programming 27 (1983) 241–262. ·Zbl 0526.90060 ·doi:10.1007/BF02591902
[21]	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, 1983) pp. 530–539. ·Zbl 0552.90059
[22]	M.J. Todd, ”Polynomial expected behaviour of a pivoting algorithm for linear complementarity and linear programming problems”, Technical Report No. 595, School of Operations Research and Industrial Engineering, Cornell University, (Ithaca, NY, November 1983),Mathematical Programming 35 (1986) 173–192. ·Zbl 0613.90094
[23]	C.A. Tovey and G. Weiss, ”A note on the volumes of random simplices”, ISyE Technical Report # J-84-8, Georgia Institute of Technology, (Atlanta, GA, June 1984).

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.