ON THE SIMPLEX ALGORITHM FOR NETWORKS ... - James B. Orlin

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James B. Orlin / On the network simplex algorithm 177 ·..:··., .i ''''` leads to a geometric improvement towards the optimal solution. Roohy-Laleh (1980) in an unpublished PhD thesis, and independently, Hung (1983) developed different polynomial time pivoting procedures for the assignment problem by exploiting property (2) above. Both relied on variants of Cunningham's method for avoiding stalling. Professor G.B. Dantzig (1983) independently proved that his simplex rule in conjunction with the strongly feasible pivot rule converges geometrically to the optimum solution. His argument is essentially the same as the one given here for the proof of Theorem 3. Although the above worst-case analyses are based on the use of Dantzig's pivot rule, we may relax this restriction substantially. For example, the same bound is valid if we consider an alternative pivot rule in which the min-variable is selected at least once every k pivots for some fixed k. Also, the same bound is valid if the entering variable x is such that kjltJ < ICiminl for some fixed k. Finally we note that the worst case analysis does not reflect the average performance of the simplex method. For example, in a recent very elegant paper, Haimovich (1983) gave a detailed probabilistic analysis of a variant of the simplex algorithm and proved that under suitable probabilistic assumptions the average number of pivots is O(n). Acknowledgments I wish to thank Professors George Dantzig, Bill Cunningham, Fred Glover, John VandeVate, and the anonymous referees for their suggestions and help. References *' ..·- ·-· "'' ` '' ' ' ' ··; · R. Barr, F. Glover and D. Klingman, "The alternating basis algorithm for assignment problems", Mathematical Programming 13 (1977) 1-13. A. Charnes, "Optimality and degeneracy in linear programming", Econometrica 20 (1952) 160-170. W.H. Cunningham, "A network simplex method", Mathematical Programming 11 (1976) 105-116. W.H. Cunningham, "Theoretical properties of the network simplex method", Mathematics of Operations Research 4 (1979) 196-208. G.B. Dantzig, Linear programming and extensions (Princeton University Press, Princeton, NJ, 1963). G.B. Dantzig, Personal communication (1983). R. Dial, F. Glover, D. Karney and D. Klingman, "A computational analysis of alternative algorithms and labeling techniques for finding shortest path trees', Networks 9 (1979) 215-248. J. Edmonds, unpublished manuscript (1970). J. Elam, F. Glover and D. Klingman, "A strongly convergent primal simplex algorithm for generalized networks", Mathematics of Operations Research 4 (1979) 39-59. G. Glover and D. Klingman, "Network optimization: New methods and applications", presented at the 10th International Symposium on Mathematical Programming (1979). M. Haimovich, "The simplex algorithm is very good! - On the expected number of pivot steps and related properties of random linear programs", unpublished manuscript (1983). ~·-% Ri9esPlfll~g4lllBB~_. PORIM---- M-~--m I, ~--~-__-___c_ -- ------ -I·
178 James B. Orlin / On the network simplex algorithm "' '' I '' "` '' "";"`" ' '' M. Hung, "A polynomial simplex method for the assignment problem", Operations Research 31 (1983) 595-600. E. Roohy-Laleh, "Improvements to the theoretical efficiency of the network simplex method", Ph.D. Dissertation (Carleton University, Ottawa, 1980). R. McBride, "Efficient solution of generalized network problems", unpublished manuscript (1981). A. Orden, "The transshipment problem", Management Science 2 (1956) 276-285. J. Orlin, "A polynomial time simplex algorithm for the minimum cost network flow problem", accepted for publication by Operations Research (1984). A.F. Veinott, Jr., "Extreme points of Leontief substitution systems", Linear Algebra and its Applications 1 (1968) 181-194. N. Zadeh, "A bad network flow problem for the simplex method and other minimum cost flow algorithms", Mathematical Programming 5 (1973) 255-266. N. Zadeh, "Near-equivalence of network flow algorithms", Technical Report 26, Department of Operations Research (Stanford University, Stanford, CA, 1979). 1L ' ·-··.·L..,·.. ""c...; -~~~~~~~~~~~~BP~~~~~~~~~~~~~~~~·1(13111111111P~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1L31~~~~~~~~~~~~~~~~~~~~~~~~~~~ II1- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 4- --* --- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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ON THE SIMPLEX ALGORITHM FOR NETWORKS ... - James B. Orlin

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