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Polynomial expected behavior of a pivoting algorithm for linear complementarity and linear programming problems.(English)Zbl 0613.90094

We show that a particular pivoting algorithm, which we call the lexicographic Lemke algorithm, takes an expected number of steps that is bounded by a quadratic in n, when applied to a random linear complementarity problem f dimension n. We present two probabilistic models, both requiring some nondegeneracy and sign-invariance properties. The second distribution is concerned with linear complementarity problems that arise from linear programming. In this case we give bounds that are quadratic in the smaller of the two dimensions of the linear programming problem, and independent of the larger.

MSC:

90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
90C05 Linear programming
68Q25 Analysis of algorithms and problem complexity
65K05 Numerical mathematical programming methods

Cite

References:

[1]I. Adler, ”The expected number of pivots needed to solve parametric linear programs and the efficiency of the self-dual simplex method”, manuscript, Department of Industrial Engineering and Operations Research, University of California, Berkeley, California (May 1983).
[2]I. Adler and S.E. Berenguer, ”Random linear programs”, Operations Research Center Report No. 81-4, University of California, Berkeley, California (1981).
[3]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”, manuscript, Department of Industrial Engineering and Operations Research, University of California, Berkeley, and Department of Computer Science, Stanford University, Stanford, California (November 1983).
[4]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 (1984), pp. 312–323.
[5]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, Illinois (April 1983).
[6]K.H. Borgwardt, ”Some distribution-independent results about the asymptotic order of the average number of steps of the simplex method”,Mathematics of Operations Research 7 (1982) 441–462. ·Zbl 0498.90054 ·doi:10.1287/moor.7.3.441
[7]K.H. Borgwardt, ”The average number of pivot steps required by the simplex-method is polynomial”,Zeitschrift fur Operations Research 26 (1982) 157–177. ·Zbl 0488.90047 ·doi:10.1007/BF01917108
[8]M. Haimovich, ”The simplex algorithm is very good! - On the expected number of pivot steps and related properties of random linear programs”, manuscript, Columbia University, New York, New York (April 1983).
[9]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
[10]J.H. May and R.L. Smith, ”Random polytopes: their definition, generation, and aggregate properties”,Mathematical Programming 24 (1982) 39–54. ·Zbl 0491.90060 ·doi:10.1007/BF01585093
[11]N. Megiddo, ”The probabilistic analysis of Lemke’s algorithm for the linear complementarity problem”, manuscript, Department of Computer Science, Stanford University, Stanford, California (September 1983).
[12]N. Megiddo, ”Improved asymptotic analysis of the average number of steps performed by the self-dual simplex algorithm”, manuscript, Department of Computer Science, Stanford University, Stanford, California (September 1983). ·Zbl 0618.90061
[13]N. Megiddo, ”On the expected number of linear complementarity cones intersected by random and semi-random rays”, manuscript, Department of Computer Science, Stanford University, Stanford, California (November 1983). ·Zbl 0613.90092
[14]T.S. Motzkin, ”The probability of solvability of linear inequalities”, in: H.A. Antosiewicz, ed.,Proceedings of the second symposium in linear programming (National Bureau of Standards and Directorate of Management Analysis, USAF, 1955) pp. 607–611.
[15]A. Prekopa, ”On the number of vertices of random convex polyhedra”,Periodica Mathematica Hungarica 2 (1972) 259–282. ·Zbl 0282.60007 ·doi:10.1007/BF02018666
[16]R. Saigal, ”On some average results for linear complementarity problems”, manuscript, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois (August 1983).
[17]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
[18]S. Smale, ”The problem of the average speed of the simplex method”, in: A. Bachem, M. Grotschel and B. Korte, eds.,Mathematical programming: the state of the art (Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1983) pp. 530–539. ·Zbl 0552.90059
[19]M.J. Todd, ”Complementarity in oriented matroids”, to appear inSIAM Journal on Algebraic and Discrete Methods. ·Zbl 0556.05016
[20]M.J. Todd, ”Linear and quadratic programming in oriented matroids”, Technical Report No. 565, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York (March 1983).
[21]L. Van der Heyden, ”A variable dimension algorithm for the linear complementarity problem”,Mathematical Programming 19 (1980) 328–346. ·Zbl 0442.90090 ·doi:10.1007/BF01581652
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.
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