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

....generation level. The results, y, of the electricity model are passed to the gas inventory problem and a new cut is created. The process is repeated until an optimal solution is found. This approach is known as Benders decomposition. A comprehensive discussion of this subject can be found in [16]. Here is a formal description of the suggested algorithm. ffl Initialization. Set the number of cuts K to zero and Psi to Gamma1. ffl General Step. 1. Determine the values of u, z, y K 1 , and Psi by solving the power generation problem, fmin GammaE(cz) Psi : u 2 U; z 2 Z(U ) g(u; z) ....

G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, editors. Optimization, volume 1 of Handbooks in Operations Research and Management Science. North-Holland, 1989.

....This fact makes problems that at first sight seem impossible (for example, control problems that one wishes to solve in something like real time) tractable. In the past twenty years rather sophisticated and reliable techniques for small scale problems have been developed (see Chapters 1 and 3 of Nemhauser et al. 1989, and the chapters of Bartholomew Biggs and Fletcher in this volume) However, efficient algorithms for small scale problems do not necessarily translate into efficient algorithms for largescale problems (see, for example, Bartholomew Biggs and Hernandez, 1994) Thus, it is not adequate to take ....

....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) and Wright (1991) The book by Mor e and Wright (1993) besides having a useful introduction to ....

G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd. Optimization, Volume 1 of Handbooks in Operations Research and Management Science. North-Holland, Amsterdam, 1989.

.... The structural enabling bound of a given transition t of N is SE(t) def = max fk j M = M 0 C Delta oe 0; oe 0 : M kPRE[t] g (LPP1) Note that the definition of structural enabling bound reduces to the formulation of a linear programming problem, that can be solved in polynomial time [NRKT89] Now let us remark the relation between behavioural and structural enabling bound concepts that follows from the implication M 0 [oeiM ) M = M 0 C Delta oe oe 0 . Property 4.3 [CCS91a] Let hN ; M 0 i be a net system. For any transition t in N , SE(t) E(t) As we remarked before, the ....

.... Gamma (j) max 8 : Y T Delta PRE Delta D (j) Y T Delta M 0 j Y T Delta C = 0 ; Y 0 9 = 19) The previous lower bound for the mean interfiring time (or its inverse, an upper bound for the throughput) can be formulated in terms of a fractional programming problem [NRKT89] and later, after some considerations, transformed into a linear programming problem. p 3 p 16 p 7 p 10 p 12 p 9 p 8 p 11 t 1 t 2 t 3 t 4 t 5 p 1 p 2 p 4 p 5 t 6 p 6 p 13 p 14 p 15 t 13 t 7 t 8 t 9 t 10 t 11 t 12 N 1 N 2 N 2 Figure 5: A live ....

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G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, editors. Optimization, volume 1 of Handbooks in Operations Research and Management Science. North-Holland, Amsterdam, The Netherlands, 1989.

....Maximize P M j=1 p k s.t. P M j=1 p k q k i = r i i =1; ###;N p j # 0 j =1; ###;M If P M j=1 p j =1is not satisfied, all p j s can be divided by this sum to obtain the desired result. Since, linear programming problems are shown to belong to the class of polynomial (P) problems (see [15] for example) rationalization stands decidable. ....

G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, editors. Optimization, volume 1 of Handbooks in operations research and management science. North-Holland, Amsterdam, 1989.

....and the net is non live. The advantage of theorem 2.1 lies on the fact that the simplex method for the solution of LPP s has almost linear complexity in practice, even if it has exponential worst case complexity. In any case, algorithms of polynomial worst case complexity can be found in [13]. For strongly connected marked graphs, the bound derived from theorem 2.1 has been shown to be reachable for arbitrary mean values and coefficients of variation associated with transition service times [5] Unfortunately, this is not the case for more general net subclasses. Let us consider, for ....

G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, editors. Optimization, volume 1 of Handbooks in Operations Research and Management Science. NorthHolland, Amsterdam, The Netherlands, 1989.

....that satisfies the axioms, well use it. If it does not, then don t. And, this provides a very good reason for the success of linear programming, i.e. there are many applications and the software that has been developed over the years provides us with very efficient solution techniques, e.g. [41]. The main algorithms were originally based on Dantzig s simplex method. Then, begun by the work of Karmarkar 1985 [31] a revolution started in the way we solve LPs and led to the p d i p methods which has resulted in a much greater increase in size and speed, e.g. 39] However, in the article ....

G. L. NEMHAUSER, A. H. G. RINNOOY KAN, and M. J. TODD, editors. Optimization, volume 1 of Handbooks in Operations Research and Management Science. North-Holland Publishing Co., Amsterdam, 1989.

....has the drawback that it generally yields an algorithm with a time and space complexity that is exponential in the number of sequences in the input. We study a new approach to solving sequence alignment problems based on an area of combinatorial optimization known as polyhedral combinatorics [34,27]. We demonstrate how this approach when applied to the Generalized Maximum Trace and RNA Sequence Alignment problems yields an algorithm for each problem that is not based on dynamic programming but is known as a branch and cut algorithm [15] Branch and cut algorithms combine linear programming ....

G. L. Nemhauser, A. Kan, H. G. Rinnooy, and M. J. Todd, editors. Optimization, volume 1 of Handbooks in Operations Research and Management Science. North-Holland, Amsterdam, 1989.

....(LPP2) The basic advantage of the above statement lies in the fact that the simplex method for the solution of LPPs has almost linear complexity in practice, even if it has exponential worst case complexity. In any case a discussion on algorithms of polynomial worst case complexity can be found in [36]. Corollary 5.1 Assuming that F i 0 and that there do not exist circuits containing only immediate transitions, the problem (LPP2) has unbounded solution iff 9Y 0; Y 6= 0 such that Y T Delta M 0 = 0 and Y T Delta C = 0. If the solution of problem (LPP2) is unbounded, since it is a lower ....

G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, editors. Optimization, volume 1 of Handbooks in Operations Research and Management Science. North-Holland, Amsterdam, 1989.

.... transition t i can be derived: Gamma (i) max Y 2fP Gammasemif lowg Y T Delta PRE Delta D (i) Y T Delta M 0 The previous lower bound can be formulated in terms of a fractional programming problem and later, after some considerations, transformed into a linear programming problem [17]: Theorem 2.1 [5] For any live and bounded net, a lower bound for the mean interfiring time Gamma (i) of transition t i can be computed by the following linear programming problem: Gamma (i) maximum Y T Delta PRE Delta D (i) subject to Y T Delta C = 0 Y T Delta M 0 = 1; ....

....The basic advantage of theorem 2.1 lies in the fact that the simplex method for the solution of a linear programming problem has almost linear complexity in practice, even if it has exponential worst case complexity. In any case, algorithms of polynomial worst case complexity can be found in [17]. For strongly connected marked graphs, the bound derived from theorem 2.1 has been shown to be reachable for arbitrary mean values and coefficients of variation associated with transition service times [3] This is not the case, in general, for live and bounded free choice nets. However, a ....

G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, editors. Optimization, volume 1 of Handbooks in Operations Research and Management Science. North-Holland, Amsterdam, The Netherlands, 1989.

.... i can be derived: Gamma (i) max Y 2fP Gammasemif lowg Y T Delta PRE Delta D (i) Y T Delta M 0 (7) The previous lower bound has been formulated in [4] in terms of a fractional programming problem and later, after some considerations, transformed into a linear programming problem [20]: Property 2.1 [4] For any live and bounded system, a lower bound for the mean interfiring time Gamma (i) of transition t i can be computed by the following linear programming problem: Gamma (i) maximum Y T Delta PRE Delta D (i) subject to Y T Delta C = 0 Y T Delta M 0 = ....

....The basic advantage of property 2.1 lies in the fact that the simplex method for the solution of a linear programming problem has almost linear complexity in practice, even if it has exponential worst case complexity. In any case, algorithms of polynomial worst case complexity can be found in [20]. In order to interpret property 2.1, let us consider again the net system of figure 1. Assuming, for instance, that all routing rates associated with output transitions at conflicts in p 1 and p 2 are equal to one, then the system (4) gives v (1) 1 ( 1 is a vector with all entries ....

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G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, editors. Optimization, volume 1 of Handbooks in Operations Research and Management Science. North-Holland, Amsterdam, The Netherlands, 1989.

....j=1 p k q k i = r i i = 1; Delta Delta Delta ; N p j 0 j = 1; Delta Delta Delta ; M If P M j=1 p j = 1 is not satisfied, all p j s can be divided by this sum to obtain the desired result. Since, linear programming problems are shown to belong to the class of polynomial (P) problems (see [15] for example) rationalization stands decidable. ....

G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, editors. Optimization, volume 1 of Handbooks in operations research and management science. North-Holland, Amsterdam, 1989.

....a few: management, control theory, physics, engineering. The goal of this paper is to give an overview on nonlinear optimization both on the theory as on practical methods, putting an emphasis on algorithms. There are already several textbooks and surveys about this field, like [1] 7] 10] [23], 38] 3] 4] All of them contain some theory and numerical methods. What is new here is the attempt to present all classes of methods used for solving optimization problems: numeric, interval or symbolic. This survey does not pretend to give a full and complete presentation of all ....

G. L. Nemhauser, A. H. G. Rinnooy Kahn, and M. J. Todd. Optimization, volume 1 of Handbooks in Operations Research and Management Science. North Holland, Amsterdam, 1989.

....with trees are of major importance in combinatorial optimization. Tree problems arise in many applications as in telecommunication network design, computer networking and facility location. For a thorough treatment of trees, applications, theoretical and algorithmic issues, see Magnanti and Wolsey [4]. In some applications one is interested in trees with additional properties, like diameter or degree constraints, or that subtrees (ooe a root node) satisfy a cardinality constraint, see [4] Recently Gouveia [3] studied the problem of nding a minimum weight spanning tree (in a given graph) ....

.... a thorough treatment of trees, applications, theoretical and algorithmic issues, see Magnanti and Wolsey [4] In some applications one is interested in trees with additional properties, like diameter or degree constraints, or that subtrees (ooe a root node) satisfy a cardinality constraint, see [4]. Recently Gouveia [3] studied the problem of nding a minimum weight spanning tree (in a given graph) satisfying hop constraints. The situation may described as follows. Let r be a given ( xed) node in a graph G. For a spanning tree T we de ne, for each v # T , dist(v) as the number of edges in ....

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T.L. Magnanti and L.A. Wolsey. Optimal trees, volume 7 of Handbooks in Operations Research and Management Science, chapter 9, pages 503615. North-Holland, 1995.

....simplex method [12] for the solution of LPP s gives good results in practice, even if it has exponential worst case complexity. Moreover, the simplex method gives feasible solutions being basic solutions. In any case, a discussion on algorithms of polynomial worst case complexity can be found in [30]. Theorem 3.1 shows that the problem of finding an upper bound for the steady state throughput (lower bound for the mean cycle time) in a strongly connected stochastic MG can be solved looking at the mean cycle time associated with each minimal P semiflow (circuits for MG s) of the net, considered ....

....in the condensation graph, and label these transitions in such a way that: 8T i ; T j : i j = T j 6 T i . Step 2: For each SCC N i , i = 1; k, considered in isolation: 1. Solve the linear programming problem (LPP2) for example, using one of the polynomial algorithms presented in [30]) Let Gamma min Gammapot (i) be its optimum value. 2. For each transition t of N i , solve the linear programming problem (LPP7) for example, using one of the polynomial algorithms presented in [30] Let be Gamma max Gammapot (i) P t ( t =SEB(t) Step 3: For each SCC N i , i = 1; ....

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G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, editors. Optimization, volume 1 of Handbooks in Operations Research and Management Science. North-Holland, Amsterdam, The Netherlands, 1989.

....scheduling algorithms and ILP based algorithms have been used. Although an ILP formulation always solves the scheduling problem optimally, care has to be taken so that the formulation can be also be solved efficiently (because solving a general ILP is an NP hard problem [2] It has been said in [5] that formulating a good model is of crucial importance to solving the model . Therefore, to efficiently solve the scheduling problem, it is important to use a structured formulation, and a mathematical analysis of the constraints is needed to find a structured formulation. Well known ILP based ....

G. L. Nemhauser and L.A. Wolsey. Optimization, volume 1 of Handbooks in Operations Research and Management Science, chapter 6. Elsevier Science Publishers B. V., 1989.

....ellipsoid method of D. B. Yudin and A. S. Nemirovskii could be applied to yield a polynomial time algorithm for linear programming, but it was not a practical method for large scale problems. These developments are well described in Dantzig s and Schrijver s books [4, 25] and the edited collection [18] on optimization. In 1985, Karmarkar [9] proposed a new polynomial time method for linear programming which did lead to practically useful algorithms, and this led to a veritable industry of developing so called interior point methods for linear programming problems and certain extensions. One ....

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

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G. L. Nemhauser, A. Kan, H. G. Rinnooy, and M. J. Todd, editors. Optimization, volume 1 of Handbooks in Operations Research and Management Science. North-Holland, Amsterdam, 1989.

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G.L. Nemhauser and A.H.G. Rinnooy Kan, editors. Stochastic Models, volume 2 of Handbooks in Operations Research and Management Science. North Holland, 1990. edited by D.P. Heyman and M.J. Sobel.

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G.L. Nemhauser and A.H.G. Rinnooy Kan, editors. Optimization, volume 1 of Handbooks in Operations Research and Management Science. North--Holland, Amsterdam, 1989. edited by G.L. Nemhauser, A.H.G. Rinnooy Kan and M.J. Todd.

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