J. F. Benders, "Partitioning procedures for solving mixed-variables programming problems," Numer. Math., vol. 4, pp. 238--252, 1962.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to row generation in contrast to column generation. The unimodular probing method described in Section 4.3 is such a row generation method. Another form of hybridization based master slave problem concept is Benders decomposition. The idea of Benders decomposition was rst presented in [Ben62] and generalized in [Geo72] Integration of Benders decomposition and constraint programming was described in [EW01] and [Tho01] The classical Benders decomposition is presented below. Consider the linear program P given by; P : min f c i y i subject to G i x A i y i b i 8i (4.6) ....

J. F. Benders. Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik, 4:238-252, 1962.

....MINLP in a Lagrangian formulation and then decomposes it into subproblems in such a way that, after fixing a subset of the variables, the resulting subproblem is convex and can be solved easily. There are three methods to implement this approach. 47 Generalized Benders decomposition (GBD) [67, 77, 29] computes at each iteration an upper bound on the solution sought by solving a primal problem and a lower bound on a master problem. The primal problem corresponds to the original problem with fixed discrete variables, while the master problem is derived through nonlinear duality theory. The ....

J. F. Benders. Partitioning procedures for solving mixed-variables programming problems. Numer. Math, pages 238--242, 1962.

....easily. Although these methods may not require the derivatives of continuous subproblems, di#erentiability can help improve solution time and quality. For this reason, we classify these methods under this class. There are three types of these methods. a) Generalized Benders decomposition (GBD) [64, 71, 44] computes at each iteration an upper bound on the solution sought by solving a primal problem and a lower bound on a master problem. The primal problem corresponds to the original problem with fixed discrete variables, whereas master problem is derived through nonlinear duality theory. Its major ....

J. F. Benders. Partitioning procedures for solving mixed-variables programming problems. Numer. Math, pages 238--242, 1962.

....have a significantly more (or less) adverse impact on future decisions. We thus need to look at the intrinsic value of each remnant piece and to do this we define a corresponding dual problem. In order to solve the problem we therefore choose a different approach, namely, Benders decomposition [5]. The basic idea is to decompose the problem into hard variables (the integer z variables) and easy variables (the continuous variables) and alternate between solving a linear program and an integer program at each iteration. To apply Benders method, we first consider the above IP but for a ....

Benders, J.F., 1962, Partitioning Procedures for Solving Mixed-Variables Programming Problems," Numerische Mathematik , 4, 238-252.

....constrained MINLP in a Lagrangian formulation before decomposing it into subproblems in such a way that after 51 fixing a subset of the variables, the resulting subproblem is convex and can be solved easily. There are three classes of these algorithms. a) Generalized Benders Decomposition (GBD) [56, 71, 21] is used to solve a subclass of constrained MINLPs under some convexity assumptions. For example, it requires the continuous subspace to be a nonempty and convex set and the objective and constraint functions to be convex. Its basic idea is to generate in each iteration an upper bound on the ....

J. F. Benders. Partitioning procedures for solving mixed-variables programming problems. Numer. Math, pages 238--242, 1962.

....The first stage consists of one inequality which specifies that the total investment cannot exceed the fixed budget. The second stage is an arc chain formulation for maximizing total flow of all commodities. A technique developed by Van Slyke and Wets [1] which in turn is based on Bender s [2] decomposition method, is used. However, it is specialized to exploit the special structure of this problem. The solution technique involves first choosing an arbitrary solution to the first stage constraint. Then the second stage program is solved to obtain an expected value for the objective ....

Benders, J. F., "Partitioning Procedures for Solving Mixed-Variables Programming Problems, Num. Math, 4, pp. 238-252, 1962.

....is convex in a subspace and can be solved easily. Although these methods may not require the di#erentiability of functions, derivative information can help improve solution time and quality significantly. There are three types of these methods. 29 Generalized Benders decomposition (GBD) [60, 68, 37] generates in each iteration an upper bound on the solution sought by solving a primal problem and a lower bound on a master problem. The primal problem corresponds to the original problem with fixed discrete variables and provides upper bound information and Lagrange multipliers for each ....

J. F. Benders. Partitioning procedures for solving mixed-variables programming problems. Numer. Math, pages 238--242, 1962.

....of the basis inverse. Indirect methods essentially follow the forkjoin protocol, by decomposing the original problem into decoupled first level subproblems and then solving a master coordination problem. Eckstein [27] surveys parallel implementations to date. The Dantzig Wolfe [19] and Benders [7] decomposition are two classic indirect methods. When applied to (CBA) both can be implemented in parallel, since each block can be assigned to a PE. In the fork phase each PE solves the associated subproblem; then, in the join, a PE collects subproblem solution information, partially redefines ....

J.F. Benders. Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik, 4:238--252, 1962.

....is convex in a subspace and can be solved easily. Although these methods may not require the di#erentiability of functions, derivative information can help improve solution time and quality significantly. There are three types of these methods. 29 Generalized Benders decomposition (GBD) [60, 68, 37] generates in each iteration an upper bound on the solution sought by solving a primal problem and a lower bound on a master problem. The primal problem corresponds to the original problem with fixed discrete variables and provides upper bound information and Lagrange multipliers for each ....

J. F. Benders. Partitioning procedures for solving mixed-variables programming problems. Numer. Math, pages 238--242, 1962.

....2.1. The Multicut L Shaped Method The L shaped method of Van Slyke and Wets [25] for solving (5) proceeds by finding subgradients of partial sums of the terms that make up Q (4) together with linear inequalities that define the domain of Q. The method is essentially Benders decomposition [2], enhanced to deal with infeasible iterates. A full description is given in Chapter 5 of Birge and Louveaux [5] We sketch the approach here and show how it can be implemented in an asynchronous fashion. We suppose that the second stage scenarios indexed by 1; 2; N are partitioned into T ....

J. F. Benders. Partitioning procedures for solving mixed variable programming problems. Numerische Mathematik, 4:238--252, 1962.

....exploits the fact that in some problems, xing the values of certain di cult variables simpli es the problem tremendously. By enumerating those di cult variables, solving each resulting subproblem and selecting the best subproblem solution found, the original problem can be solved. Benders method [1] is more ingenious. It solves a master problem to assign values to the di cult variables. Each solution to the subproblem then generates a Benders cut that is added to the master problem before resolving it. Thus each solution to the master problem must satisfy all the Benders cuts obtained so ....

....0; q 2 Q 2 ; 4) x 2 D; where u q is the dual solution to the subproblem when the subproblem is feasible in iterations q 2 Q 1 , and infeasible in iterations q 2 Q 2 , before resolving the master problem in the next iteration. A more detailed description of Benders Decomposition can found in [1, 8, 11, 14, 17]. 2.2 Previous Integration Schemes Properties of a number of di erent problems were considered by Darby Dowman and Little in [4, 5] and their e ect on the performance of CLP and MIP approaches were presented. They reported experimental results that illustrate some key properties of the ....

J. F. Benders. Partitioning procedures for solving mixed-variables programming problems. Numer. Math., 4:238252, 1962.

....between 3.1 3.8 on 4 processors are achieved. Re optimization with the Primal Dual Interior Point Method 1 1 Introduction A number of optimization algorithms require solving a sequence of linear programs. This is a common situation, for example, in the cutting plane methods [12] decomposition [5, 3], branch and bound and branch and cut approaches for mixed integer optimization [23] and many others. The problems in such a sequence are often similar, i.e. the following instance is only a minor perturbation of the earlier one. Hence the optimal solution of the earlier problem (or, more ....

....such as the coe#cient matrix, the right hand side, the objective function and or the variable bounds do change. This is the case for example when subproblems in Dantzig Wolfe decomposition [5] are solved (the objective of the linear program changes) when subproblems in Benders decomposition [3] are solved (the right hand side vector of the linear program changes) or when a variable has its bound tightened in the branch and bound technique (this again results in the perturbation of the right hand side vector) Preliminary results of applying an interior point based reoptimization ....

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J. F. Benders, Partitioning procedures for solving mixed-variables programming problems, Numerische Mathematik, 4 (1962), pp. 238--252.

....easily. Although these methods may not require the derivatives of continuous subproblems, differentiability can help improve solution time and quality. For this reason, we classify these methods under this class. There are three types of these methods. a) Generalized Benders decomposition (GBD) [64, 71, 44] computes at each iteration an upper bound on the solution sought by solving a primal problem and a lower bound on a master problem. The primal problem corresponds to the original problem with fixed discrete variables, whereas master problem is derived through nonlinear duality theory. Its major ....

J. F. Benders. Partitioning procedures for solving mixed-variables programming problems. Numer. Math, pages 238--242, 1962.

....a branch and bound algorithm can be guaranteed only if the bounding step generates valid upper and lower bounds on the mixedinteger nonconvex problem. 2.1.1 Upper bound A rigorous upper bound is obtained by solving the nonconvex MINLP (1) locally. The generalized Benders decomposition (GBD) (Benders, 1962; Geoffrion, 1972; Floudas et al. 1989) may be used to obtain such a solution. When there are no mixed bilinear terms, the outer approximation with equality relaxation (OA ER) Duran and Grossmann, 1986; Kocis and Grossmann, 1987) or a local MINLP branch and bound algorithm (B B) Beale, 1977; ....

Benders, J. F., "Partitioning Procedures for Solving Mixed-Variables Programming Problems, " Numer. Math., 4, 238 (1962).

....and considerable computing facilities. Many real life linear programs (LPs) display some particular block structures which deserve special treatment by an optimization code. A well known example is the block angular structure of constraints that can be handled by a decomposition approach [4, 11]. Decomposition allows tight control of memory requirements and considerable computational e#ciency, in particular through parallel computation [5] An alternative is to use a direct solution method. There are two major approaches to the solution of linear programs: simplex method [10] and ....

J. F. Benders, Partitioning procedures for solving mixed-variables programming problems, Numerische Mathematik, 4 (1962), pp. 238--252.

....Global constraints can be viewed as equivalent to a set of conditional constraints. These ideas are developed further in [85, 87, 91] 4.1. 4 Benders Decomposition Another promising framework for integration is a logic based form of Benders decomposition, a well known optimization technique [16, 59]. The variables are partitioned (x, y) and the problem is written, minimize f(x) subject to h(x) g i (x, y) all i (11) The variable x is initially assigned a value x that minimizes f(x) subject to h(x) This gives rise to a feasibility subproblem in the y variables: g i (x, y) all i 23 ....

Benders, J. F., Partitioning procedures for solving mixed-variables programming problems, Numerische Mathematik 4 (1962): 238-252.

....of dom L. Dantzig Wolfe decomposition is thus equivalent to Lagrangian relaxation. It can also be extended to the case of X not convex, for instance by including integrality restrictions, as in first outlined in Gilmore and Gomory [30, 31] 4 2.3. Benders decomposition. Benders decomposition [7] deals with the problem min f(x) g(y) h(x) k(y) 0; x 2 X; y 2 Y; where X ae R n is an arbitrary set, Y ae R p is convex, f : X 7 R and g : Y 7 R are convex, h : R n 7 R m is convex and k : Y 7 R m is convex. Finally, for the sake of simplicity, we assume that g; h and k are ....

J. F. Benders, "Partitioning procedures for solving mixed-variables programming problems", Numerische Mathematik 4 (1962), 238-252.

....THE PROBLEM WITH BENDERS DECOMPOSITION Since the number of variables p j in problem (8) can grow exponentially with the number of atomic propositions, just as in ordinary probabilistic logic, we need again to use a column generation technique. This can be done by using Benders decomposition [2] to isolate the p j s in a subproblem, to which column generation is applied. The nonlinear constraints remain in the master problem. In our example, the master problem is, max P r(x 3 ) s.t. 4) u r u r 0 0; r = 1; R (9) 0 P r( Delta) 1 (10) where is a vector of ....

Benders, J. F., Partitioning procedures for solving mixed variables programming problems, Numerische Mathematik 4 (1962) 238-252.

....the circuits are equivalent. The distinguishing feature of this algorithm is that each input enumerated gives rise to a class of inputs that need not be enumerated. Precisely the same strategy is used by Benders decomposition, a well known technique for solving mathematical programming problems [2, 18]. In fact we discovered our algorithm by formulating the circuit verification problem as an integer programming problem and solving it with Benders decomposition. We found that the resulting algorithm is formally equivalent to the nonnumeric algorithm we describe here. For this reason we refer to ....

....Decomposition 9 inequalities equivalent to C and D, then (c d) T x (fl ffi ) is a clausal inequality equivalent to E, and fx j c T x fl; d T x ffi; 0 x eg = fx j (c d) T x (fl ffi ) 0 x eg: 1. 6 Interpretation as Benders Decomposition To apply Benders decomposition [2, 18] to the circuit verification problem we must write it as an integer programming problem. We first write each equation y i = f i (y J i ) as a conjunction of clauses. To do this we use the fact that a logical equivalence A j B can be written ( A B) A :B) Recall that f i (y J i ) is written in ....

Benders, J. E. Partitioning procedures for solving mixed-variables programming problems. Numerical Mathematics 4 (1962) 238-252.

....computation. We have recently developed a highly parallel formulation for sparse linear programming problems using interior point methods [11] Nevertheless, dense linear programming problems have some genuine applications. For instance the Dantzig Wolfe decomposition [4] or Benders decomposition [1] generate highly dense master problems. More applications 1 leading to dense linear programming problems are discussed in [5] Furthermore, in problems in which the nonzero elements are a fixed percentage d of the total number of elements, the formulations presented here lead to scalable ....

J. F. Benders. Partitioning procedures for solving mixed variable programming problems. Numerische Mathematik, (4):238--252, 1962.

....optimization problems and their formulation as integer programs. Then we will review a general representation theory for integer programs that gives a formal measure of the expressiveness of this algebraic approach. We conclude this section with a representation theorem due to Benders [10] which has been very useful in solving certain large scale combinatorial optimization problems in practice. 3.1 Formulations Formulating decision problems as integer or mixed integer programs is often considered an art form. However, there are a few basic principles which can be used by a novice ....

....MILP representable if and only if it is the union of finitely many polyhedra having the same set of recession directions. 17 3.3 Benders Representation Any mixed integer linear program (MILP) can be reformulated so that there is only one continuous variable. This reformulation, due to Benders [10], will in general have an exponential number of constraints. Benders representation suggests an algorithm for mixed integer programming (known as Benders Decomposition in the literature because of its similarity to Dantzig Wolfe Decomposition, cf. 95] that uses dynamic activation of these rows ....

J.F.Benders, Partitioning procedures for solving mixed-variables programming problems, Numerische Mathematik 4 (1962), pp. 238-252.

....those vast resources. This is not caused by the lack of appropriate parallel algorithms: those have been proliferating for more than a decade now (not to mention the parallel methods that came before the time of parallel computers) One can enumerate decomposition based approaches like [DW60, Ben62, RW91, Bir85, MR95, Rus93, HS91, R#97] data parallel algorithms [Rus95, KRS93, KR94] and even specializations of general optimization methods for solution of a structured problem, This research was a part of Special Research Program SFB F011 Aurora supported by Austrian Fonds zur F#rderung der ....

....methods below. 4.2 The selected methods 4.2. 1 The Dantzig Wolfe and Benders decomposition methods A number of methods, including some of the oldest and best known, were based on the famous decomposition principle of Dantzig and Wolfe [DW60, Dan63] Although the so called Benders decomposition [Ben62] is typically seen as dual to Dantzig Wolfe approach, we shall try to express them both using the same terminology. Let us rst recall the notion of a subgradient of convex function. The subdioeerential (a set of subgradients) of convex function f at x (0) is dened as f(x (0) fa : 8x ....

J.F. Benders. Partitioning procedures for solving mixedvariable programming problems. Numerische Matematik, 4:238252, 1962.

.... and a (k) 2 c (k) c(x (k) where the generalized gradient (or subdi erential) is de ned as f(x) conv n g j g = lim i 1 rf(x i ) x i x; rf(x i ) exists converges o Problems of type (P ) arise as master problems in decomposition methods such as Benders Decomposition (e.g. [2], 10] and [9] An important evolving area of application Numerical Analysis Report NA 195, Department of Mathematics, University of Dundee y Department of Mathematics, University of Dundee, DD1 4HN, U.K. fletcher maths.dundee.ac.uk, sleyffer maths.dundee.ac.uk 1 2 R. Fletcher and S. ....

J.F. Benders. Partitioning procedures for solving mixed-variable programming problems. Numerische Mathematik, 4:238-252, 1962.

....to analyze the cost tradeoff between centralization and decentralization capacity planning. Further Discussions and Remarks There are different decomposition techniques available to solve the stochastic models developed in this section (see Ruszczynski (1997) among which Benders decomposition (Benders (1962)) is the most widely used one due to its simplicity and effectiveness. Application of Benders decomposition to solve the models in a decentralized environment can be interpreted as follows. The center announces a capacity configuration and expansion decision vector. The manufacturing and ....

Benders, J.F., 1962, "Partitioning Procedures for Solving Mixed Variables Programming Problems", Numer. Math., 4:238-252.

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J. F. Benders. Partitioning procedures for solving mixed variables programming problems. Numerische Mathematik, 4:238-252, 1962.

....few components. This research was partially supported by U.S. Oce of Naval Research Grant N0001495 1 0517 and by the Engineering Design Research Center at Carnegie Mellon University, an Engineering Research Center of the National Science Foundation, under grant EEC8943164. Benders decomposition [7, 17] uses a problem solving strategy that can be generalized to a larger context. It assigns some of the variables trial values and nds the best solution consistent with these values. In the process it learns something about the quality of other trial solutions. It uses this information to reduce the ....

Benders, J. F., Partitioning procedures for solving mixed-variables programming problems, Numerische Mathematik 4 (1962) 238-252.

....equivalent problem: x p(#)d # y # Ax = b T# x W# y # = h# 0, y # (3) Date: 17 januari 2003. The structure of equation T# x W# y # = h# is called L shaped, which can be made visible by writing it out: 4) TK x WK y K = hK 2. Benders Algorithm The Benders algorithm[1, 6] to solve this problem can be formulated as follows (we largely follow the notation in[5] Step 1: Initialization # : 1 Iteration number UB : # Upper LB : # Lower Solve initial master problem: Ax = b (5) Step 2: Sub problems for # ## do Solve the sub problem: min d ....

J. F. Benders, Partitioning procedures for solving mixed-variables programming problems, Numerische Mathematik 4 (1962), 238--252.

....small rectangular bounds. For some specific classes of BMI problems, for example the low order controller design problem, heuristic specialized methods with local convergence have been developed [BG96] The method presented in this paper uses a technique called generalized Benders decomposition [Ben62, Geo72]. It can be interpreted as an extension of the GOP algorithm of Visweswaran and Floudas[FV93, VFIP96] which is a global optimization method for bilinear and biconvex optimization problems. The outline of the paper is as follows. In the next section we present the duality theory, which provides ....

J. F. Benders. Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik, 4:238--252, 1962.

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J. F. Benders, "Partitioning procedures for solving mixed-variables programming problems," Numer. Math., vol. 4, pp. 238--252, 1962.

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J.F. Benders. Partitioning procedures for solving mixed variables programming problems. Numerische Mathematik, 4, 1962.

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Benders, J.F., "Partitioning procedures for solving mixed variables programming problems", Numerische Mathematik 4, (1962), pp 238-252.

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J.F. Benders. Partitioning procedures for solving mixedvariable programming problems. Numerische Matematik, 4:238252, 1962.

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J.F. Benders. Partitioning procedures for solving mixed-variable programming problems. Numer. Math., 4:238-252, 1962.

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J. F. Benders. Partitioning procedures for solving mixed-variable programming problems. Numer. Math., 4:238-252, 1962.

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J.F. Benders. Partitioning procedures for solving mixed{variable programming problems. Numerische Matematik, 4:238-252, 1962.

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J. F. Benders. Partitioning procedures for solving mixed-variable programming problems. Numer. Math., 4:238-252, 1962.

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Benders, J.F. (1962): Partitioning Procedures for Solving Mixed-Variable Programming Problems, Numerische Mathematic 4, 238-252.

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J. F. Benders, 1962. Partitioning procedures for solving mixed-variables programming problems, Numerische Mathematik 4, 238-252.

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J. F. Benders. Partitioning procedures for solving mixed variable programming problems. Numerische Mathematik, 4:238--252, 1962.

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J. Benders, Partitioning procedures for solving mixed-variables mathematical programming problems. Numersche Mathematik, 4 (3): 238-252, 1960.

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Benders, J.F. (1962). Partitioning Procedures for Solving Mixed-Variables Programming Problems. Numerische Mathematic 4, 238-252.

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J. F. Benders. Partitioning procedures for solving mixedvariables programming problems. Numerische Mathematik, 4:238--252, 1962.

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J. F. Benders (1962), "Partitioning procedures for solving mixed-variable programming problems", Numerische Mathematik 4, 238-252.

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Benders, J. (1960), "Partitioning procedures for solving mixed-variables mathematical programming problems," Numersche Mathematik , 4 (3): 238-252.

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Benders J.F. (1962), Partitioning Procedures for Solving Mixed-Variables Programming Problems, Numerische Mathematik, vol. 4, 238-252.

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Benders J.F. (1962), Partitioning Procedures for Solving Mixed-Variables Programming Problems, Numerische Mathematik, vol. 4, 238-252.

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J.F. Benders. Partitioning procedures for solving mixed-variable programming problems. Numerische Mathematik, 4:238--252, 1962.

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Benders J.F. (1962), Partitioning Procedures for Solving Mixed-Variables Programming Problems, Numerische Mathematik, vol. 4, 238-252.

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Benders J.F. (1962), Partitioning Procedures for Solving Mixed-Variables Programming Problems, Numerische Mathematik, vol. 4, 238-252.

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Benders J.F. (1962), Partitioning Procedures for Solving Mixed-Variables Programming Problems, Numerische Mathematik, vol. 4, 238-252.

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