J.R. Birge. Decomposition and partitioning methods for multistage stochastic linear programs. Operations Research, 33:9891007, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....times. For this purpose, most of the solution methods in the area are based on specialized decomposition; we refer to [10] and the references therein for a survey along this direction. A classical approach in that case is the so called L shaped method [27] and its multistage extension [7], which are variants of the Benders decomposition. Recently, multistage linear stochastic programs with millions of variables and constraints have been solved with parallel implementations of Benders decomposition [8, 14, 16] However, Benders decomposition is limited to problems with a linear ....

J.R. Birge, Decomposition and partitioning methods for multistage stochastic linear programs, Operations Research 33, 989-1007, 1985.

....with linear or convex quadratic stochastic programs as special cases. This paper is concerned with the ecient numerical treatment of such inherently large scale problems when stochastic in uences are modeled by a scenario tree. Well known numerical approaches include primal decomposition methods [4, 10, 15], dual decomposition methods [16, 20] and interior point methods [2, 8, 23] for a more exhaustive overview see [5, 22] In any case the key to success lies in taking advantage of the characteristic problem structure. This is achieved by decomposition into node or scenario subproblems (primal and ....

J. R. Birge, Decomposition and partitioning methods for multistage stochastic linear programs, Oper. Res., 33 (1985), pp. 989-1007.

....using any of the algorithms described above. Multistage programs, however, are computationally much more challenging, as they suffer from the well known curse of dimensionality . Algorithms for multistage programs have been developed, but not many numerical studies are presently available. Birge [3] generalized the L shaped decomposition method of Van Slyke and Wets [28] to multistage programs, and Gassmann [11] provides experimentation with an 2 implementation of this algorithm. The progressive hedging algorithm of Rockafellar and Wets [24] is also applicable to the multistage case. ....

....T , and the inner product x T y is written xy when context makes the meaning clear. Bold letters are used to denote stochastic quantities, and the corresponding roman letters designate instances of these quantities. A T stage stochastic programming problem can be formulated as follows (Birge [3]) MS] min n c 1 x 1 E 2 h min i c 2 x 2 E 3 j 2 i min c 3 x 3 Delta Delta Delta E T j 2 ; T Gamma1 min c T x T jjio x 1 x 2 x 3 x T s.t. A 1 x 1 = b 1 ; B 2 x 1 A 2 x 2 = b 2 ; B 3 x 2 A 3 x 3 = b 3 ; B T x T Gamma1 A T x T = b T ; 0 x t ....

J.R. Birge. Decomposition and partitioning methods for multistage stochastic linear programs. Operations Research, 33(5):989--1007, 1985.

....at forklare til ikke matematikere. 3. I forbindelse med dekompositionsalgoritmer er scenarie problemerne nemme at lse med specielt effektive netvaerksalgoritmer. Generalisation til flere trin To trins SP modellen kan naturligt generaliseres til et Multistage Stochastic Program , her med T trin [2]: min n c 1 x 1 E 2 h min i c 2 x 2 E 3 j 2 i minc 3 x 3 Delta Delta Delta E T j 2 ; T Gamma1 minc T x T jjio x 1 x 2 x 3 x T s.t. A 1 x 1 = b 1 ; B 2 x 1 A 2 x 2 = b 2 ; B 3 x 2 A 3 x 3 = b 3 ; B T x T Gamma1 A T x T = b T ; 0 x t u t ; for t = ....

J.R. Birge. Decomposition and partitioning methods for multistage stochastic linear programs. Operations Research, 33(5):989--1007, 1985.

....use the simplex method or interior point method to solve a linear program (see (2.9) equivalent to (1.1) 1.3) cf [6] 7] 8] 20] 36] ii) cutting plane based decomposition methods (CPDM) which generate a set of cuts to approximate the nonlinear and nonsmooth problem (1. 1) cf [1] 2] [3] [5] 10] 37] 38] and (iii) derivative based decomposition methods (DDM) which determine a search direction by using the gradient (and Hessian) at the current point and find a new point along this direction, cf [25] 28] 29] 30] 31] 32] Generally speaking, each method has advantages and ....

Birge, J.R. (1985): Decomposition and partitioning methods for multi-stage stochastic linear programs. Operations Research, 33, 989--1007. 30

....context makes the meaning clear. Bold letters are used to denote stochastic quantities, and the corresponding roman letters designate instances of these quantities. 4. 1 Formulation of the T stage Stochastic Program A T stage stochastic programming problem can be formulated as follows (Birge [2]) MS] min n c 1 x 1 E 2 h min i c 2 x 2 E 3 j 2 i min c 3 x 3 Delta Delta Delta E T j 2 ; T Gamma1 min c T x T jjio x 1 x 2 x 3 x T s.t. A 1 x 1 = b 1 ; 24 B 2 x 1 A 2 x 2 = b 2 ; B 3 x 2 A 3 x 3 = b 3 ; B T x T Gamma1 A T x T = b T ; 0 ....

J.R. Birge. Decomposition and partitioning methods for multistage stochastic linear programs. Operations Research, 33(5):989--1007, 1985.

....decision. Addressing the realistic requirement that there should be more than two stages, so that decisions at any (but the last) stage are still made under further uncertainty, leads to multistage SPs, MSP. A T stage stochastic programming problem can be formulated as follows (Birge [3]) min x1 c 1 x 1 E 2 [min x2 (c 2 x 2 E 3 j 2 (min x3 c 3 x 3 E T j 2 ; T 1 min xT c T x T ) s.t. A 1 x 1 = b 1 ; B 2 x 1 A 2 x 2 = b 2 ; B 3 x 2 A 3 x 3 = b 3 ; B T x T 1 A T x T = b T ; 0 x t u t ; for t = 1; T ; where t = A t ; B t ; b ....

Birge, J.R.: `Decomposition and Partitioning Methods for Multistage Stochastic Linear Programs ', Operations Research 33, no. 5 (1985), 989{ 1007.

...., and the inner product x T y is written xy when context 12 makes the meaning clear. Bold letters are used to denote stochastic quantities, and the corresponding roman letters designate instances of these quantities. A T stage stochastic programming problem can be formulated as follows (Birge [1]) MS] min n c 1 x 1 E 2 h min i c 2 x 2 E 3 j 2 i min c 3 x 3 Delta Delta Delta E T j 2 ; T Gamma1 min c T x T jjio x1 x2 x3 x T s.t. A 1 x 1 = b 1 ; B 2 x 1 A 2 x 2 = b 2 ; B 3 x 2 A 3 x 3 = b 3 ; B T x T Gamma1 A T x T = b T ; 0 x t u t ; for ....

J.R. Birge. Decomposition and partitioning methods for multistage stochastic linear programs. Operations Research, 33(5):989--1007, 1985.

....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 ....

....reformulations of existing sequential algorithms, which allow exploiting more parallelism (see, e.g. HM92] 2. New inherently parallel algorithms were designed. a) Some are based on a rather coarse grain parallel structure closely related to the problem domain. Node decomposition on a tree [Bir85, Rus93] or scenario decomposition [RW91, MR95] may serve as examples. b) Others exploit only the ne grain structure of the constraint matrix of problem (2) and are thus well suited to solve both structured and unstructured problems [Rus95, KR94] It is possible that the run time of some of ....

[Article contains additional citation context not shown here]

J.R. Birge. Decomposition and partitioning methods for multistage stochastic linear programs. Operations Research, 33:9891007, 1985.

....programming paradigms widely used today are task parallelism and data parallelism. While quite some e ort has been made in order to develop task parallel methods for both unstructured and structured large scale sparse optimization problems (for some of the more recent references see, e.g. [3, 4, 9, 12, 26, 27, 33, 35, 37, 11, 17]) until recently little has 12 been known about data parallel approaches to those problems. Works [34, 15, 14] presented a family of parallel methods for convex optimization problems, including a specialization for linear programs. The algorithms exhibit the following common characteristics: ....

J.R. Birge. Decomposition and partitioning methods for multistage stochastic linear programs. Operations Research, 33:9891007, 1985.

....Lagrangian. 16 Figure 10: Scenario tree example from Figure 8 expanded into separate scenarios linked by nonanticipativity constraints. 1 1 1 1 1 2 1 2 5 6 3 4 4 4 7 8 9 10 6. 3 Nested Benders or regularized decomposition Nested Benders (possibly regularized) decomposition [Ben62, Bir85, Rus93b] is another node oriented decomposition method for solution of our tree structured problem (1) At each nonterminal node n a synchronization function Q(x n ) is appended to the original objective. It represents the lower bound on the cost to go , i.e. the expected objective value increase after ....

....one stage to another and synchronizes between the stages. There are many possibilities of directing the ow of currently solved node subproblems, however certain amount of synchronization is always required. Nested regularized decomposition [Rus93b] develops ideas of nested Benders decomposition [Bir85] and two stage regularized decomposition [Rus86] Unlike its predecessors, it allows asynchronous parallel execution of both master and slave problems at all nodes of the tree, thus greatly diminishing the scalability concerns caused by the need to synchronize for information passing. A complete ....

J.R. Birge. Decomposition and partitioning methods for multistage stochastic linear programs. Operations Research, 33:9891007, 1985.

....to the most difficult problems of mathematical programming. Their solution requires either simulation based methods, such as stochastic quasi gradient methods [2] if one deals with a general distribution of the random parameters, or special decomposition methods for very large structured problems [1, 8, 14, 15], if the distribution is approximated by finitely many scenarios. Most of the existing computational methods (such as, e.g. all decomposition methods) are applicable only to convex problems. The methods that can be applied to multi extremal problems, like some stochastic quasi gradient methods of ....

Birge J.R. (1985), Decomposition and partitioning methods for multistage stochastic linear programs, Operations Research 33(1985) 989-1007.

....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, R97] 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 ....

....reformulations of existing sequential algorithms, which allow exploiting more parallelism (see, e.g. HM92] 2. New inherently parallel algorithms were designed. a) Some are based on a rather coarse grain parallel structure closely related to the problem domain. Node decomposition on a tree [Bir85, Rus93] or scenario decomposition [RW91, MR95] may serve as examples. b) Others exploit only the ne grain structure of the constraint matrix of problem (2) and are thus well suited to solve both structured and unstructured problems [Rus95, KR94] It is possible that the run time of some of ....

[Article contains additional citation context not shown here]

J.R. Birge. Decomposition and partitioning methods for multistage stochastic linear programs. Operations Research, 33:9891007, 1985.

....discretizations of the probability space and possibly the time horizon. Nevertheless, the characteristic structure of scenario trees makes these problems tractable by numerical algorithms. Among the most prominent ones are several variants of decomposition methods. Primal decomposition approaches [4, 8, 11] assign a small local optimization problem to every node and treat the vertical coupling between stages iteratively, by passing intermediate solutions up and objective and feasibility cuts down the scenario tree. 2 M. C. Steinbach Tree Recursion and Successive Refinement in MSP Dual ....

J. R. Birge. Decomposition and partitioning methods for multistage stochastic linear programs. Oper. Res., 33(5):989--1007, 1985.

....to solve the problem. Even more important are bounding procedures in multi stage problems. In the last decade several authors, such as Birge, Gassman, Higle, Sen, Louveaux, Ruszczynski and Wets, developed various solution techniques for multi stage stochastic programming problems (see e.g. [1], 5] 6] 2] 3] 7] These methods are capable to solve problems at most 4 5 stages, because the size of the stochastic programming problem exponentially increases with the number of stages. This is called the curse of dimensionality in dynamic programming. Thus, bounding procedures may ....

Birge J.R. (1985). Decomposition and partitioning methods for multi-stage stochastic linear programs, Operations Research, 33, 989--1007

....the branches. The constraints T n x p(n) A n x n = b n which de ne the dynamics of the decision process, also link the subproblems. They are relaxed by placing them in the augmented Lagrangian. 6. 3 Nested Benders or regularized decomposition Nested Benders (possibly regularized) decomposition [Ben62, Bir85, Rus93b] is another node oriented decomposition method for solution of our tree structured problem (1) At each nonterminal node n a synchronization function Q(x n ) is appended to the original objective. It represents the lower bound on the icost to goj, i.e. the expected objective value increase after ....

....one stage to another and synchronizes between the stages. There are many possibilities of directing the AEow of currently solved node subproblems, however certain amount of synchronization is always required. Nested regularized decomposition [Rus93b] develops ideas of nested Benders decomposition [Bir85] and two stage regularized decomposition [Rus86] Unlike its predecessors, it allows asynchronous parallel execution of both master and slave problems at all nodes of the tree, thus greatly diminishing the scalability concerns caused by the need to synchronize for information passing. A complete ....

J.R. Birge. Decomposition and partitioning methods for multistage stochastic linear programs. Operations Research, 33:9891007, 1985.

....discretizations of the probability space and possibly the time horizon. Nevertheless, the characteristic structure of scenario trees makes these problems tractable by numerical algorithms. Among the most prominent ones are several variants of decomposition methods. Primal decomposition approaches [3, 6, 9] assign a small local optimization problem to every node and treat the vertical coupling between stages iteratively, by passing intermediate solutions up and objective and feasibility cuts down the scenario tree. Dual decomposition and progressive hedging algorithms [13, 15, 17] optimize ....

J. R. Birge, Decomposition and partitioning methods for multistage stochastic linear programs, Oper. Res., 33 (1985), pp. 989--1007.

....the next section we show its relevance to the efficiency of the solver. 4. From model to solution We apply SET to produce dual block angular structures from stochastic linear programs formulated with GAMS modeling language. Dual block angular structure is well suited to the Benders decomposition [2,20,27]. The decomposition method we use belongs to this class. The Analytic Center Cutting Plane 11 Figure 5. Dual block angular structure Figure 6. Moving columns from the second stage to the first stage Figure 7. Resulting dual block angular structure Method solves the restricted master problem and ....

J. R. Birge, Decomposition and partitioning methods for multistage stochastic linear programs, Operations Research, 33 (1985), pp. 989--1007.

....Griffith University, Nathan, QLD 4111. y School of Computer and Information Technology, Griffith University, Nathan, QLD 4111. restore feasibility at some cost. Most of the methods used are variants of the L shaped method[11] This method has been extended to multistage methods by Birge [1]. In this paper a technique known as the progressive hedging method [8] is tested for its parallization properties.This algorithm was successfully applied to portfolio management by Mulvey and Vladimirou. A quadratic program has to be solved in each iteration of progressive hedging algorithm for ....

J.R. Birge. Decomposition and partitioning methods for multistage stochastic linear programs. Oper. Res., 33:989, 1985.

....AA T matrix [27] Finally, let us mention the whole variety of more complicated structures that, in particular, can be the result of nested embedding of some of the basic structures described earlier. A classical example of such a structure arises in multi stage stochastic programming problems [4]. Despite their variety, the above mentioned structures share a common property: all of them can be described by a block partitioning of the constraint matrix. Let A 2 R m Thetan be the block partition representation of the constraint matrix A = 2 6 6 6 6 6 6 4 A 00 A 01 : A 0N A 10 A ....

J. R. Birge, Decomposition and partitioning methods for multistage stochastic linear programs, Operations Research, 33 (1985), pp. 989--1007.

....the following properties: i) X is adapted to the filtration fF t g. ii) x 1 satisfies (2.1) and x 1 0. iii) X satisfies (2.2) and is nonnegative P a.s. Computing a strong solution for multi stage stochastic linear programs is the subject of a large number of research papers, see e.g. Birge [3], Birge and Wallace [5] Dantzig and Infanger [9] Entriken and Infanger [11] Gassman [12] Higle and Sen [13] Infanger [17] Rockafellar and Wets [24] Ruszczynski [25] and Wets [28] Unfortunately, when the number of possible realizations of the uncertain parameters is large, efficient ....

Birge J.R., (1985), Decomposition and Partitioning Methods for Multistage Stochastic Linear Programs, Operations Research 33, 989--1007.

.... Secondly, we have a number of special decomposition methods which exploit the structure of the problem to split it into manageable pieces and coordinate their solution [23] One can distinguish two classes: primal decomposition methods that work with subproblems which are assigned to time stages [4, 7, 17, 18, 22] and dual methods, in which subproblems correspond to scenarios [11, 19, 16] In this paper we shall use the general theory of augmented Lagrangian decomposition of [20] to develop and analyze two new decomposition methods for multistage stochastic programs. The first one is a dual method ....

J.R. Birge, "Decomposition and partitioning methods for multistage stochastic linear programs", Operations Research 33(1985) 989-1007.

....be found in a series of pioneer papers of Rockafellar and Wets [16] 17] 18] 19] 21] 22] Algorithms for linear quadratic cases of problem (1.1) in which both OE(x) and (y) are linear or quadratic, have been studied extensively. Among them are the L shaped method [26] the decomposition methods [1][5] the finite generation method [9] 21] the projected gradient method [32] the steepest descent method [31] the SQP method [14] and some interior point methods [2] 3] 23] 28] 29] For the special case where both X and Y are boxes and both OE(x) and (y) are separable quadratic functions, a ....

J. R. Birge, "Decomposition and partitioning methods for multistage stochastic linear programs", Operations Research, 33(1985)989-1007.

....S ( x Gamma ) ffl) X. Chen Stochastic Programming 9 6 Final remarks This paper provides a brief overview of recent results on Newton type methods for solving two stage linear stochastic programs with fixed recourse. These results can be extended to multistage stochastic programming problems [2, 19, 27]. For instance, we can employ smooth approximation in some stages, and obtain a smooth approximation function to the original problem. The other interesting issue is to study some algorithms combining Newton type methods with other methods, for example, the stochastic decomposition method [4, 15, ....

J.R. Birge, Decomposition and partitioning methods for multistage stochastic linear programs, Oper. Res. 33 989-1007 (1985).

.... Secondly, we have a number of special decomposition methods which exploit the structure of the problem to split it into manageable pieces and coordinate their solution [27] One can distinguish two classes: primal decomposition methods that work with subproblems which are assigned to time stages [4, 8, 21, 22, 26] and dual methods, in which subproblems correspond to scenarios [13, 23, 19] In this paper we shall use the general theory of augmented Lagrangian decomposition of [24] to develop and analyze two new decomposition methods for multistage stochastic programs. The first one is a dual method ....

J.R. Birge, "Decomposition and partitioning methods for multistage stochastic linear programs", Operations Research 33(1985) 989-1007.

....The solution of the subproblem is then used in the main problem to generate a better approximate solution. There are basically two kinds of methods of solving the main problem. One is the cutting plane method which approximates the main problem through an increasingly refined set of cuts [2] [3] [9] 27] 28] The other kind finds iterates through search directions which are usually determined by derivatives of the objective and constraint functions in the main problem, 18] 21] 22] 23] 24] 25] One of the major differences between the direct and indirect methods lies in their ....

J.R. Birge, "Decomposition and partitioning methods for multi-stage stochastic linear programs," Operations Research, 33 (1985) 989--1007.

....programs, the amount of time that is needed for those recursive algorithms grows up exponentially with the number of stages. Methods for solving the multi stage stochastic program, which are not based on recursive solutions of two stage programs, can be found in the works of Birge [9] and [10], Rockafellar and Wets [77] Ruszczynski [81] and others. Many of the above algorithms for solving the two stage problem use the optimal solution of Q(x 1 ; for all (or for a sample of 1 ; N ) In those algorithms, the use of parallel processing can speed them up. The use of ....

Birge J.R., (1985), Decomposition and Partitioning Methods for Multistage Stochastic Linear Programs, Operations Research 33, 989--1007. REFERENCES 115

....the algebraic modeling language. Second, SPI passes the special structure of the problem to a SES (Structure Exploiting Solver) SET can be integrated with any Algebraic Modeling Language (AML) as shown in Figure 3. The model tested in this section is a simple financial planning model suggested by Birge (1985). Possibilities of investment at each period consist of 4 securities and cash. There are no transaction costs. The initial capital and the desired wealth at the end of PSfrag replacements USER DATA MODEL RESULTS AML RESULTS DICTIONARY MP SOLUTION SET SPI SES Fig. 3. Integrating SET with an AML 0 ....

Birge, J. R. (1985). Decomposition and partitioning methods for multistage stochastic linear programs.

....cutting plane decomposition. Cutting plane decomposition was first proposed for two stage stochastic programs by Van Slyke and Wets [37] Its multicut variant is attributed to Ruszczynski [36] and Birge and Louveaux [7] while its extension to multistage stochastic programs is due to Birge [3]. Recent empirical assessments of their computational performance on various computer systems can be found in [1, 6, 12, 35, 42] not to mention the numerous studies that apply parallel algorithms based on variable splitting and augmented Lagrangian principles, or parallel computations in matrix ....

....decomposition. As we pointed it out, the subproblems in the second stage inherit the block structure typical of stochastic programming. If the subproblems become large, it would be natural to solve them by decomposition. This would lead to the use of a nested decomposition such as for example [3, 19, 32] Acknowledgments The authors gratefully acknowledge the thorough work of the referees and the editor. Their constructive criticisms led to a major revision of the paper and made it possible to greatly improve the presentation. ....

J. R. Birge, Decomposition and partitioning methods for multistage stochastic linear programs, Operations Research, 33 (1985), pp. 989--1007.

....which then solves the current linear programming approximation for new pricing information which is then transmitted to the subproblems; the latter reacts to these new prices with revised proposals, etc. There exists a variety of decomposition methods applicable to stochastic optimization problems [3, 18, 28, 29]. In this paper we are concerned with the decomposition approach that uses interior point methods. We underline our interest in decomposition and not in a direct application of an interior point method that was also proved to be an interesting alternative [5, 25] Consider the problem maximize hc ....

....generated by GAMS. Then we present the execution times to solve these problems on a cluster of 10 LINUX PC s. We also give some comments on speed up in a parallel implementation. 4. 1 Description of the model The model considered in this section is a simple financial planning problem suggested in [3] (It is just used here for its generic properties and definitely not for its modeling content) At each period, one can invest in 4 securities and cash. There are no transaction costs. The initial capital and the desired wealth at the end of the planning period are defined exogenously. Prices of ....

J. R. Birge, Decomposition and partitioning methods for multistage stochastic linear programs, Operations Research, 33 (1985), pp. 989--1007.

....In order to generate trees with a number of scenarios other than 2 i , a part of binary trees was cutted. The program MSLiP [Gas90] is the version MSLiP 8.3, version of April 7, 1995. Gas90] This is a general purpose program for solving MultiStage Linear Programs using the L shaped Method [Bir85]. Therefore, the comparison is made with respect to the question, how much is the advantage of using an adapted algorithm instead of a general purpose algorithm. MSLiP consider nonstochastic objective functions only, thus additional variables were introduced. The examples were computed on a ....

J.R. Birge. Decomposition and partitioning methods for multistage stochastic linear programs. Operation Research, 33:989--1007, 1985.

....is poor compared to the simplex algorithm and due this reason all implementations have used only the simplex method to solve both the master and sub problems of the L shaped method. Developing a warm start capability for the interior point method is active area of research at present. Birge ([5]) extended the L shape method to Multi stage stochastic linear programs with recourse and referred to this algorithm as Nested Decomposition Algorithm. 2.1.1 Parallelisation A key observation of this method is that there are a large number of sub problems, and that they are independent. ....

J.R. Birge. Decomposition and partitioning methods for multistage stochastic linear programs. Operations Research, 33, 1985.

....and incurring additional costs. The overall objective is to minimize costs at time 1 plus expected costs at time 2. There exist several solution methods for two stage stochastic linear problems: specializations of the simplex method [8, 9, 13] different variants of decomposition schemes [2, 3, 24, 30, 31, 32] and other computationally attractive approaches [26, 29, 34] The progress in interior point methods (IPMs) for linear programming [19, 14] makes them also attractive candidates, worth to be tried. An advantageous feature of IPMs is that they converge very rapidly, in 20 to 50 iterations, almost ....

Birge J. (1985) Decomposition and partitioning methods for multistage stochastic linear programs, Operations Research 33, pp. 989-1007.

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J.R. Birge, 1985. Decomposition and partitioning methods for multi--stage stochastic linear programs, Operations Research 33, 989-1007.

....models to capture hedging behavior can lead to poor planning (see, for example, Birge [9] and costly future penalties. One example of solving a unit commitment problem under uncertainty, involving pump storage, is in Pereira and Pinto [31] who used a method similar to that given in Birge [7]. Our basic model follows these developments. To ease our development here, we assume that the power flow equations and other relationships can be linearized (as in De ak et al. 18] or Granville and Pereira [20] In this case, the stochastic version of P(3) becomes a multistage stochastic mixed ....

....pump storage and thermal commitment decisions by developing solution patterns corresponding to feasible cycles of varying sets of units. Each pattern y i , i 2 I, corresponds to fixing some set of integer variable levels within each x t;i . I This allows us to decompose the problem as in Birge [7], Louveaux [27] and Pereira and Pinto [31] and enables parallel computation. Techniques such as lagrangian relaxation also allow bounds on the solution values and determine when the solution process can stop. We give the details of our approach in the following section. In this procedure, we ....

J.R. Birge. Decomposition and partitioning methods for multi--stage stochastic linear programs. Operations Research 33:. 989-1007, 1985.

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J.R. Birge. Decomposition and partitioning methods for multistage stochastic linear programs. Operations Research, 33:9891007, 1985.

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J.R. Birge. Decomposition and partitioning methods for multistage stochastic linear programs. Operations Research, 33:9891007, 1985.

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Birge J.R. (1985), Decomposition and Partitioning Methods for Multistage Stochastic Linear Programs, Operational Research, vol. 33, 989-1007.

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Birge J.R. (1985), Decomposition and Partitioning Methods for Multistage Stochastic Linear Programs, Operational Research, vol. 33, 989-1007.

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Birge, J.R. (1985), Decomposition and Partitioning Methods for Multi-Stage Stochastic Linear Programs, Operations Research, vol. 33, 989-1007.

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Birge J.R. (1985), Decomposition and Partitioning Methods for Multistage Stochastic Linear Programs, Operational Research, vol. 33, 989-1007.

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Birge, J.R., 1984, "Decomposition and Partitioning Methods for Multistage Linear Programs", Operations Research, Vol.33, No.5.

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Birge J.R. (1985), Decomposition and Partitioning Methods for Multistage Stochastic Linear Programs, Operational Research, vol. 33, 989-1007.

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J. R. Birge, Decomposition and partitioning methods for multistage stochastic linear programs, Operations Research, 33 (1985), pp. 9891007.

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Birge J.R., (1985), Decomposition and Partitioning Methods for Multistage Stochastic Linear Programs, Operations Research 33, 989--1007.

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<F3.734e+05> J. R.<F3.821e+05> Birge,<F3.467e+05> Decomposition and partitioning methods for multistage stochastic linear<F3.821e+05> programs, Oper. Res., 33 (1985), pp. 989--1007.

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Birge, J.R. (1985) Decomposition and partitioning methods for multistage stochastic linear programs, Opns. Res. 33, 989-1007.

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