D. Goldfarb and M. J. Todd. Linear programming. In G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, editors, Optimization, volume 1 of Handbooks in Operations Research and Management Science. Elsevier/North Holland, Amsterdam, The Netherlands, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....A fundamental property of linear programs is that, if the optimal objective value is finite, a vertex minimizer must exist, i.e. a point where n constraints with linearly independent gradients hold with equality. For details about linear programming and its terminology, see, e.g. 4] 31] and [19]. The simplex method, invented by George B. Dantzig in 1947, is an iterative procedure for solving LPs that depends totally on the property just mentioned. Beginning with a vertex, every iteration of the simplex method moves to an adjacent vertex, decreasing the objective as it goes, until an ....

....of interior methods typically involve barrier functions or their properties, such as perturbed complementarity 3. The Revolution Begins 7 (2. 17) Readers interested in Karmarkar s method should consult his original paper [21] or any of the many comprehensive treatments published since 1984 (e.g. [19, 30, 32, 42]) Beyond showing the formal connection between Karmarkar s method and barrier methods, 15] reported computational results comparing a state of the art (in 1985) simplex code, MINOS [27] and an implementation of the primal Newton barrier method on a widely available set of test problems. To the ....

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D. Goldfarb and M. J. Todd (1989). Linear programming, in Optimization (G. L. Nemhauser, A. H. G. Rinnooy Kan and M. J. Todd, eds.), North Holland, Amsterdam and New York, 73--170.

....to propose algorithms for large scale problems and report numerical results only for problems in a few hundred variables. Besides the relevant chapters in this volume, very good background reading in linear, constrained and unconstrained nonlinear programming is provided in the chapters of Goldfarb and Todd (1989), Dennis and Schnabel (1989) and Gill and Murray (1989) in the book by Nemhauser et al. 1989) Recent articles and books devoted primarily to large scale optimization include Coleman and Li (1990) Coleman (1993) Conn et al. 1989) Conn et al. 1990b) Conn et al. 1992b) Conn et al. 1992g) ....

D. Goldfarb and M. J. Todd. Linear programming. In G.L. Nemhauser, A.H.G. Rinnooy Kan, and M.J. Todd, editors, Optimization, Volume 1 of Handbooks in Operations Research and Management Science, pages 73--170. NorthHolland, Amsterdam, The Netherlands, 1989.

....[13] in 1984, initiated a tremendous amount of research in polynomial time methods for LP, and gave rise to many new and efficient interior point methods (IPMs) for LP. The bibliography of Kranich [15] contains more than 1300 papers on the subject. For a survey we refer to Goldfarb and Todd [3], Gonzaga [7] den Hertog [10] den Hertog and Roos [11] Todd [27] For reports on numerical efficiency of these methods we mention Lustig, Marsten and Shanno [16] McShane, Monma and Shanno [17] Mehrotra [19] The importance of Dikin s approach is that in all polynomial time IPMs for LP, ....

Goldfarb, D. and Todd, M.J. (1989), Linear Programming, in: Handbooks in Operations Research and Management Science, Optimization, 1, 141--170, G.L. Nemhauser, A.H.G. Rinnooy Kan and M.J. Todd, eds., Amsterdam. 17

....ellipsoid containing its feasible region to start the process. However, it is generally believed that in order to apply interior point methods to the same combinatorial optimization problem one needs to have the explicit listing of all of the inequalities in the LP formulation, see [GLS88] and [GT89]. For instance, Goldfarb and Todd in their survey article on linear programming write: it appears that its [Karmarkar s new algorithm] theoretical implications are far more limited than those of the ellipsoid method. Indeed, Karmarkar s algorithm requires the linear programming problem to ....

....find the maximum stable set for the corresponding graphs in polynomial time. It is common belief that in contrast to the ellipsoid method, interior point methods require explicit knowledge of the facets of the polytope on which we wish to optimize, see for instance [GLS88] and the quotation from [GT89] in the introduction. However, we can use polynomial time interior point 27 methods to optimize over STAB G in the special cases mentioned above, even though the number of facets in such polytopes may be exponentially large. In fact, the ground breaking work of Nesterov and Nemirovskii implies ....

D. Goldfarb and M. J. Todd. Linear Programming. In G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, editors, Optimization, Handbooks in Operations Research and Management Sciences. North--Holland, 1989.

....; c 2 ] iff the following weight formula is satisfiable: V F2KB (bwf 1 (F ) bwf 2 (F ) bwf 1 ( GjA) c 1 ; c 2 ] bwf 2 ( GjA) c 1 ; c 2 ] The former proves a) while the latter shows b) 2 Proof of Theorem 4.5. We apply some basic results from linear programming (cf. especially [58, 10, 64, 25]) to the characterization of tight logical consequence given in Theorem 4.4. Without loss of generality, we prove the claim for d = d 1 . Let LP denote the minimization instance of (3) By Theorem 4.4, the optimal value of LP is given by d 1 . We now transform LP into an equivalent linear program ....

D. Goldfarb and M. J. Todd. Linear programming. In G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, editors, Optimization, volume 1 of Handbooks in Operations Research and Management Science, chapter 2, pages 73--170. Elsevier Science, 1989. 54 INFSYS RR 1843-00-01

....the fact that it was possible to implement his algorithm with reasonable efficiency. The theoretical computational complexity of interior point methods for LP was eventually lowered to O( p nL) iterations, requiring a total of O(n 3 L) bit operations by a number of authors. Goldfarb and Todd [8] provide a good reference for these complexity results. By using fast matrix multiplication techniques, the complexity estimates can be reduced further. Quite recently, Anstreicher [2] proposed an interior point method, combining partial updating with a preconditioned gradient method, that has an ....

D. Goldfarb and M. J. Todd. Linear programming. In Optimization, pages 73--170. North-Holland, Amsterdam, 1989.

....Z 1 Z 2 , we maintain Z 1 , the top m 1 rows of Z, and three smaller arrays: the dense representation for V = GammaT Gamma1 R 5 and packed representations for A 2 and Q = T Gamma1 . All the necessary information will be created using the following equation whose validity is proven in [14]. Z = B Gamma1 A = B Gamma1 A 1 A 2 = Z 1 Z 2 = Z 1 V Z 1 QA 2 Because we update Z 1 in every iteration, d 1 is readily available. If ff is in Z 1 , then it is also available without further computation. So, the only vectors we need to compute are d 2 and ff ....

D. Goldfarb and M. Todd, 1989. Linear Programming. In Nemhauser, Rinnooy Kan and Todd, editors, Optimization, Handbooks in OR and MS. Vol:1, North-Holland, Amsterdam.

....provided by Thinking Machines Mario Bourgoin. 2 Data Parallel Tableau Simplex Methods Consider a linear program in the standard primal form minimize c x such that Ax = b x 0 ; 1) where A in an m Theta n matrix. Each iteration of the tableau based primal simplex method for (1) see e.g. [9, 17]) takes the following form, assuming that the reduced costs are stored in row zero of the (m 1) Theta (n 1) tableau T , and the right hand side entries in column zero: T1. Select Entering Variable: Choose a pivot column j 0 such that t 0j 0. The simplest form of the method locates ....

....contexts. 2. 3 Computational results with the tableau code On CM 2 hardware configurations with varying numbers of processors and local memory capacities, we tested a code similar to the stripe array implementation just outlined, but incorporating optional upper bounds on variables (see e.g. [9, 17] for the techniques used) and using the EXPAND rule of Gill et al. 13] in place of the naive ratio test described above. This latter modification, which is based on a pivot row selection approach originally proposed by Harris [18] was necessary to produce acceptable numerical behavior. However, ....

[Article contains additional citation context not shown here]

D. Goldfarb and M. Todd, 1989. Linear Programming, Optimization, G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd (eds.), Handbooks in Operations Research and Management Science, Volume 1. North-Holland, Amsterdam.

....technique. We shall illustrate its power when discussing approximation algorithms. We shall also talk about network flow algorithms where linear programming plays a crucial role both algorithmically and combinatorially. For a more in depth coverage of linear programming, we refer the reader to [1, 4, 7, 8, 5]. A linear program is the problem of optimizing a linear objective function in the decision variables, x 1 : x n , subject to linear equality or inequality constraints on the x i s. In standard form, it is expressed as: Min n X j=1 c j x j (objective function) subject to: n X j=1 a ....

D. Goldfarb and M. Todd. Linear programming. In Handbook in Operations Research and Management Science, volume 1, pages 73--170. Elsevier Science Publishers B.V., 1989.

....ellipsoid containing its feasible region to start the process. However, it is generally believed that in order to apply interior point methods to the same combinatorial optimization problem one needs to have the explicit listing of all of the inequalities in the LP formulation, see [32] and [27]. For instance, Goldfarb and Todd in their survey article on linear programming write: it appears that its [Karmarkar s new algorithm] theoretical implications are far more limited than those of the ellipsoid method. Indeed, Karmarkar s algorithm requires the linear programming problem to be ....

....thus find the maximum stable set for the corresponding graphs in polynomial time. It is common belief that in contrast to the ellipsoid method, interior point methods require explicit knowledge of the facets of the polytope on which we wish to optimize, see for instance [32] and the quotation from [27] in the introduction. However, we can use polynomial time interior point methods to optimize over STAB G in the special cases mentioned above, even though the number of facets in such polytopes may be Interior Point Semidefinite Programming 31 exponentially large. In fact, the ground breaking ....

D. Goldfarb and M. J. Todd, Linear Programming, in Optimization, G. L. Nemhauser, A. H. G. R. Kan, and M. J. Todd, eds., Handbooks in Operations Research and Management Sciences, North--Holland, Amsterdam, 1989.

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D. Goldfarb and M. J. Todd. Linear programming. In G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, editors, Optimization, volume 1 of Handbooks in Operations Research and Management Science. Elsevier/North Holland, Amsterdam, The Netherlands, 1989.

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D. Goldfarb and M. Todd. Linear programming. In Handbook in Operations Research and Management Science, volume 1, pages 73--170. Elsevier Science Publishers B.V., 1989.

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D. Goldfarb and M. J. Todd. Linear programming. In G. L.NemhauvIE A. H. G. Rinnooy Kan, and M. J. Todd, editors, Optimization,voluz 1 ofHandb ooks in Operations Research and Management Science. Elsevier/North Holland, Amsterdam, The Netherlands, 1989.

No context found.

D. Goldfarb and M. J. Todd. Linear programming. In G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, editors, Optimization, volume 1 of Handbooks in Operations Research and Management Science. Elsevier/North Holland, Amsterdam, The Netherlands, 1989.

No context found.

D. Goldfarb and M. J. Todd. Linear programming. In G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, editors, Optimization, volume 1 of Handbooks in Operations Research and Management Science. Elsevier/North Holland, Amsterdam, The Netherlands, 1989.

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D. Goldfarb and M.J. Todd (1989). "Linear programming," In G.L. Nemhauser, A.H.G. Rinnooy Kan, and M.J. Todd, eds., Optimization, 73-170, North Holland, Amsterdam and New York.

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Goldfarb, D. and Todd, M.J. (1989), Linear Programming, in: Handbooks in Operations Research and Management Science, Optimization, 1, 141--170, G.L. Nemhauser, A.H.G. Rinnooy Kan and M.J. Todd, eds., Amsterdam.

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Goldfarb D. and Todd M.J., Linear Programming, in: Handbooks in Operations Research and Management Science, Vol. 1, Optimization, North-Holland, Amesterdam, 1989.

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D. Goldfarb and M. J. Todd. Linear programming. In G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, editors, Optimization, volume 1 of Handbooks in Operations Research and Management Science. Elsevier/North Holland, Amsterdam, The Netherlands, 1989.

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D. Goldfarb and M.J. Todd, "Linear programming," in Handbooks in Operations Research and Management Science, Vol.I: Optimization, A.R. Kan and M.J. Todd, eds., North-Holland, Amsterdam, 1989, Chapter 2.

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D. Goldfarb and M.J. Todd. Linear programming. In G. L. Nemhauser, A.H.G. Rinnooy Kan, and M.J.Todd, editors, Optimization, volume 1 of Handbooks in Operations Research and Management Science. Elsevier/North Holland, Amsterdam, The Netherlands, 1989.

No context found.

D. Goldfarb and M. J. Todd (1989), `Linear programming', in Optimization (G. L. Nemhauser, A. H. G. Rinnooy Kan and M. J. Todd, eds.), North Holland (Amsterdam and New York), 73--170.

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