J.R. Birge and L. Qi. Computing block-angular Karmarkar projections with applications to stochastic programming. Management Science, 34(12):14721479, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... approaches that consider di#erent points x simultaneously. Asynchronous variants of the L shaped and # trust region methods are described in Sections 2.2 and 4, respectively. Other parallel algorithms for stochastic programming have been described by Birge et al. 3,4] Birge and Qi [6], Ruszczynski [21] and Frangiere, Gondzio, and Vial [9] In [3] the focus is on multistage problems in which the scenario tree is decomposed into subtrees, which are processed independently and in parallel on worker processors. Dual solutions from each subtree are used to construct a model of ....

....and in parallel on worker processors. Dual solutions from each subtree are used to construct a model of the first stage objective (using an L shaped approach like that described in Section 2) which is periodically solved by a master process to obtain a new first stage iterate. Birge and Qi [6] describe an interior point method for two stage problems, in which the linear algebra operations are implemented in parallel by exploiting the structure of the two stage problem. However, this approach involves significant data movement and does not scale particularly well. In [9] the ....

J. R. Birge and L. Qi. Computing block-angular karmarkar projections with applications to stochastic programming. Management Science, 34:1472--1479, 1988.

....2.3 Related Work Parallel stochastic optimization algorithms have been investigated during the past 15 years. In [4] an algorithm related to ATR (but without the trust region and asynchrony features) is applied to multistage problems, and implemented on a cluster of modest size. An earlier paper [5] describes an interior point approach in which the linear algebra computations are implemented in parallel, but this approach does not scale well to a large number of processors. A small PC cluster is used in [tt] where the approach is to use interior point methods for the second stage problems ....

Birge, J. R. and L. Qi, "Computing block-angular Karmarkar projections with applications to stochastic programming," Management Science 34 (1988), pp. 1472-1479.

....(say, less than fifty) of iterative steps to reach a high precision solution, almost regardless of the size of the problem. This property is certainly of critical importance in solving large scale stochastic programming problems. Moreover, the IPMs are suitable to handle nonlinear problems. In [11] Birge and Qi showed how decomposition can be achieved based on Karmarkar s original interior point method for two stage stochastic linear programming. Within the interior point method realm, in fact, two types of decomposition methods have appeared. The first type, including [11] exploits the ....

....problems. In [11] Birge and Qi showed how decomposition can be achieved based on Karmarkar s original interior point method for two stage stochastic linear programming. Within the interior point method realm, in fact, two types of decomposition methods have appeared. The first type, including [11], exploits the structure of two stage stochastic linear programming which is viewed as large size linear programming; see [21, 9, 12] for serial algorithms and [18, 29, 13] for parallel implementations. The second type of interior point decomposition methods typically specializes some of the ....

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J.R. Birge, and L. Qi, Computing block-angular Karmarkar projections with applications to stochastic programming, Management Science 34, 1472-1479, 1988.

....method, say the simplex method, to solve (2DSLP ) and in the process of implementing the method we make use of the special structure of (2DSLP ) and decompose the 4 computation of the data as much as possible. Another good example of this type is Birge and Qi s decomposition algorithm, [5], based on Karmarkar s original method for linear programming. Our method to be introduced is in this category as well. A typical scenario based method is based on the so called Lagrangian multiplier approach, or, the augmented Lagrangian multiplier approach; see [9] The key observation is that, ....

J.R. Birge, and L. Qi, Computing block-angular Karmarkar projections with applications to stochastic programming, Management Science 34, 1472-1479, 1988.

....different points x simultaneously. Asynchronous vari 4 Jeff Linderoth, Stephen Wright ants of the L shaped and 1 trust region methods are described in Sections 2.2 and 4, respectively. Other parallel algorithms for stochastic programming have been devised by Birge et al. 4] Birge and Qi [6], and Frangi ere, Gondzio, and Vial [8] In [4] the focus is on multistage problems in which the scenario tree is decomposed into subtrees, which are processed independently and in parallel on worker processors. Dual solutions from each subtree are used to construct a model of the firststage ....

....a model of the firststage objective (using an L shaped approach like that described in Section 2) which is periodically solved by a master process to obtain a new candidate first stage solution x. Parallelization of the linear algebra operations in interiorpoint algorithms is considered in [6], but this approach involves significant data movement and does not scale particularly well. In [8] the second stage problems (3) are solved concurrently and inexactly by using an interior point code. The master process maintains an upper bound on the optimal objective, and this bound along with ....

J. R. Birge and L. Qi. Computing block-angular Karmarkar projections with applications to stochastic programming. Management Science, 34:1472--1479, 1988. Jeff Linderoth, Stephen Wright: Stochastic Programming on a Computational Grid

....The theoretical analysis leads to natural solution algorithms which combine a dynamic recursion with local projections for local constraints and a Schur complement approach for global constraints, giving linear complexity in the tree size. Other interior approaches for stochastic programs include [2, 6, 9, 11, 13, 20, 23] (twostage LP case) and [5, 12, 19, 27] linear or convex multistage case) We compare our framework with these approaches and with the generalized linear quadratic control formulations developed by Rockafellar [24, 25] and Rockafellar and Wets [26] The material is organized as follows. After ....

....= f0g [ S(0) f0g [ L; b) the objective is linear; c) all constraints are formulated as dynamics or nonnegativity constraints. Under the full rank condition (A1.2 impl ) on P j , assumptions (A1.1 impl ) A2 impl ) will hold. Several interior methods have been developed for this problem class [2, 6, 9, 11, 13, 20, 23]; most of them turn out to be encompassed within our framework. TREE SPARSE CONVEX PROGRAMS 17 TABLE 1. Corresponding matrix blocks in Birge and Holmes [9, x3.4] and Steinbach [32, x4.2] index 2 indicates blocks after projection) given generated [9] A 0 W l T l D 2 0 A 0 A 0 D 2 l S ....

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J. R. BIRGE AND L. QI, Computing block-angular Karmarkar projections with applications to stochastic programming, Management Sci., 34 (1988), pp. 1472--1479.

....and then take advantage of them. For example, simplex solvers look for network constraints while interior point solvers detect dense columns. There are several papers that address the issues of structure exploitation in the implementation of interior point methods. A nonexhaustive list includes: [6, 9, 15, 16, 18, 19, 25, 26]. These papers describe specialized algorithms each exploiting one particular structure of the constraint matrix. Some of them [16, 19, 25] present dedicated parallel implementations of these methods. Another approach is to incorporate in a solver a set of routines that can support any structure. ....

....to ship all commodities [24] The problem can also be viewed as a two stage stochastic programming problem, with an investment decision in the first stage and routing decisions in the second stage. There exist specialized implementations of interior point methods for two stage stochastic problems [6, 19] that use the Sherman Morrison Woodbury formula to take advantage of the dual block angular structure of the LP constraint matrix. We will solve three di#erent classes of network optimization problems: Multicommodity flow problem, Multicommodity flow survivable network design, ....

J. R. Birge and L. Qi, Computing block-angular Karmarkar projections with applications to stochastic programming, Management Science, 34 (1988), pp. 1472--1479.

.... in which the block diagonal part of the matrix is used to eliminate many of the variables, and a preconditioned conjugate gradient method is applied to the remaining Schur complement [Castro, 1998] Techniques for stochastic programming (two stage linear problems with recourse) are described in [Birge and Qi, 1988] and [Birge and Loueaux, 1997, Section 5.6] 3 SIMPLE EXTENSIONS OF THE PRIMAL DUAL APPROACH The primal dual algorithms of the preceding section are readily extended to convex quadratic programming (QP) and monotone linear complementarity (LCP) both classes being generalizations of linear ....

Birge, J. R. and Qi, L. (1988). Computing blockangular karmarkar projections with applications to stochastic programming. Management Science, 34:1472--1479.

....method and its variants, based on the simplex method, are very popular. With the rapid growth and development in interior point methods in recent years (cf. 16] for various survey articles on interior point methods) this traditional approach to stochastic programming needs to be reconsidered. In [4] Birge and Qi showed how decomposition can be achieved based on Karmarkar s original interior point method for two stage stochastic linear programming. A few other interior point based approaches have been suggested so far in the literature; see e.g. 3, 5, 12] Zhao [20] proposed a method in ....

J.R. Birge and L. Qi, Computing block-angular Karmarkar projections with applications to stochastic programming, Management Science 34, 1472-1479, 1988.

....with finite supports. The reader may refer to [4] for a detailed survey. These algorithms can be roughly classified into three classes: i) direct methods which directly 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 ....

....costing O(rm 3 n 3 ) Hence, K = O(rm 3 ln fi Gamma1 0 n 3 ) To achieve a good bound on K for a direct method is more involved, because variables x and y are coupled in the constraints. A few works have been carried out on exploiting special structures of stochastic programs, e.g. [7] [36] The estimation of N is, on the other hand, easy in direct interior point methods, but hard in ours. The rest of this section is devoted to the estimation of N . At each complete iteration, the algorithm performs an outer iteration which updates the parameter by a factor fl, i.e. k 1 = fl ....

Birge, J.R., Qi, L. (1988): Computing block-angular Karmarkar projections with applications to stochastic programming. Management Science, 34, 1472--1479.

....by Austrian Fonds zur F#rderung der wissenschaftlichen Forschung. y Institut f#r Statistik, Operations Research und Computerverfahren, Universit#t Wien, Universit#tsstr. 5, A1090 Wien, Austria. E mail adresses: georg.pflug univie.ac.at and swietanowski bigfoot.com, respectively. 1 like [Str80, BQ90] Some generic parallel optimization algorithmic paradigms have also been developed a relatively long time ago (see, e.g. BT89, CZ97] The authors believe that one of the causes for the slow development of practical parallel optimization systems is the diOEculty of implementing even a ....

....even a conceptually simple and inherently parallel method using the parallel programming tools of today. In fact, it is immediately apparent to the reader of most of the works listed above, that a parallel implementation was only mentioned as a possible future course of research (e.g. BQ90] or that some sequential implementation was produced and simulations of parallel execution were performed (e.g. KRS93] and [Rus93] Eventually, after years of hard work new publications appear in which successful truly parallel implementations are described (e.g. BH92, VZ97, DT95, ....

[Article contains additional citation context not shown here]

J.R. Birge and L. Qi. Computing block-angular Karmarkar projections with applications to stochastic programming. Management Science, 34(12):14721479, 1990.

....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 and L. Qi. Computing block-angular Karmarkar projections with applications to stochastic programming. Management Science, 34(12):14721479, 1990.

....a large amount of excellent reviews, monographes and even textbooks. We shall therefore refer the reader to [Wri97] for a thorough introduction of IPMs, while we shall only remark brie y on one aspect of e ciency in the context of pension fund asset liability management problem. See also, e.g. [BQ90, BH92, Gas91, CFM94, Rus93a, BMV91] and references therein for more information on IPMs in the context of stochastic programming. Thanks to a special structure of our problem (1) a primal dual IPM may be successfully used for solving rather large problem instances. In IPMs the e ciency of a symmetric (possibly semi or quazi ) ....

J.R. Birge and L. Qi. Computing block-angular Karmarkar projections with applications to stochastic programming. Management Science, 34(12):1472 1479, 1990.

.... in which the block diagonal part of the matrix is used to eliminate many of the variables, and a preconditioned conjugate gradient method is applied to the remaining Schur complement [Castro, 1998] Techniques for stochastic programming (two stage linear problems with recourse) are described in [Birge and Qi, 1988] and [Birge and Loueaux, 1997, Section 5.6] 3 SIMPLE EXTENSIONS OF THE PRIMAL DUAL APPROACH The primal dual algorithms of the preceding section are readily extended to convex quadratic programming (QP) and monotone linear complementarity (LCP) both classes being generalizations of linear ....

Birge, J. R. and Qi, L. (1988). Computing blockangular karmarkar projections with applications to stochastic programming. Management Science, 34:1472--1479.

....by Austrian Fonds zur F rderung der wissenschaftlichen Forschung. y Institut f r Statistik, Operations Research und Computerverfahren, Universit t Wien, Universit tsstr. 5, A1090 Wien, Austria. E mail adresses: georg.pflug univie.ac.at and swietanowski bigfoot.com, respectively. like [Str80, BQ90] Some generic parallel optimization algorithmic paradigms have also been developed a relatively long time ago (see, e.g. BT89, CZ97] The authors believe that one of the causes for the slow development of practical parallel optimization systems is the di culty of implementing even a ....

....even a conceptually simple and inherently parallel method using the parallel programming tools of today. In fact, it is immediately apparent to the reader of most of the works listed above, that a parallel implementation was only mentioned as a possible future course of research (e.g. BQ90] or that some sequential implementation was produced and simulations of parallel execution were performed (e.g. KRS93] and [Rus93] Eventually, after years of hard work new publications appear in which successful truly parallel implementations are described (e.g. BH92, VZ97, DT95, ....

[Article contains additional citation context not shown here]

J.R. Birge and L. Qi. Computing block-angular Karmarkar projections with applications to stochastic programming. Management Science, 34(12):14721479, 1990.

....a large amount of excellent reviews, monographes and even textbooks. We shall therefore refer the reader to [Wri97] for a thorough introduction of IPMs, while we shall only remark brieAEy on one aspect of eOEciency in the context of pension fund asset liability management problem. See also, e.g. [BQ90, BH92, Gas91, CFM94, Rus93a, BMV91] and references therein for more information on IPMs in the context of stochastic programming. Thanks to a special structure of our problem (1) a primal dual IPM may be successfully used for solving rather large problem instances. In IPMs the eOEciency of a symmetric (possibly semi or quazi ) ....

J.R. Birge and L. Qi. Computing block-angular Karmarkar projections with applications to stochastic programming. Management Science, 34(12), 1990.

....P0 P1 P2 P3 P4 P5 P1 P2 P3 P4 P5 Figure 10: Dantzig Wolfe decomposable structure extracted from MERGE. P0 P1 P2 P3 P4 P5 P1 P2 P3 P4 P5 Figure 11: Row and column bordered structure extracted from MERGE. same structure can also be exploited by a specialized interior point algorithm, see e.g. [5]. Therefore, there is a sharp difference in the respective roles of SPI and SES. Although the two components are intimately linked in any application of SET, SPI is the same for all SES, while SES has to be adapted to the problem at hand. Our goal is just to make sure that the structural ....

J. R. Birge and L. Qi, Computing block-angular Karmarkar projections with applications to stochastic programming, Management Science, 34 (1988), pp. 1472--1479.

....of the equivalent large scale linear program and (b) those based on cutting planes (or equivalently on decomposition) that implicitly exploit that structure. Algorithms of class (b) are in general preferred. In 1988, following the work of Karmarkar for general linear programs, Birge and Qi [10] proposed a specialization of Karmarkar s algorithm for two stage stochastic linear programs. The algorithm of Birge and Qi [10] is the first interior point analog of class (a) Several other authors have studied related and different interior point analogs of class (a) Birge and Qi [10] also ....

....implicitly exploit that structure. Algorithms of class (b) are in general preferred. In 1988, following the work of Karmarkar for general linear programs, Birge and Qi [10] proposed a specialization of Karmarkar s algorithm for two stage stochastic linear programs. The algorithm of Birge and Qi [10] is the first interior point analog of class (a) Several other authors have studied related and different interior point analogs of class (a) Birge and Qi [10] also presented an analysis of the computational complexity of their algorithm. This analysis indicates that the computational complexity ....

[Article contains additional citation context not shown here]

J. R. Birge and L. Qi, Computing block-angular Karmarkar projections with applications to stochastic programming, Management Science 34 (1988) 1472-- 1479.

....constraint matrix into blocks associated to different subproblems. We exploit this structure to simplify the sparsity pattern analysis when building the adjacency matrix. Let us mention that several authors have already addressed the issue of the structure exploitation within the context of IPMs [3, 4, 21, 30]. The paper is organized in the following way. In Section 2 we discuss the particular structure of the LP constraint matrix in the master problem. A structure exploiting representation of the inverse of A ThetaA T matrix is subject of Section 3. The method presented in this paper has been ....

Birge J. and L. Qi, (1988). Computing block-angular Karmarkar projections with applications to stochastic programming, Management Science 34, No 12, pp. 1472-1479.

....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 simplex activeset method has been developed [20] The more general convex case of (1.1) however, has not yet received enough attention in algorithmic development although ....

J. R. Birge and L. Qi, "Computing block-angular Karmarkar projections with application to stochastic programming", Management Sciences, 34(1988)1472-1479.

....parallel. Stochastic linear programs also have diagonal blocks, but are linked together by columns of the first stage variables (linking columns) This property also has been exploited for parallelisation [23, 2, 14, 13] Jessup, Yang and Zenios [14] and Yang and Zenios [13] uses the BQ algorithm [3] to solve the sysmetric positive definite system of equations in parallel. Excellent performance is reported by them. In this paper we present a more general parallel algorithm for linear programs that have both linking rows and linking columns. Lustig and Li s [19] method is algebraically similar ....

J.R. Birge and L. Qi. Computing block-angular karmarkar projections with applications to stochastic programming. Management Science, 34:12:1472--1479, 1988.

....by using the simplex method or the interior point method. There are general purpose commercial softwares of these methods but these may not be effective. Hence many specially designed methods exploiting advantages of the special structure of these linear programs have been proposed, cf. 6] [7] [8] 14] Other methods solve the problems indirectly by using certain decomposition techniques. These methods decompose the original problem into a main problem and a subproblem (the subproblem usually comprises many small independent linear programs) Given an approximate solution to the main ....

J.R. Birge and L. Qi, "Computing block-angular Karmarkar projections with applications to stochastic programming," Management Science, 34 (1988) 1472--1479.

....of Table 3 in [2, p. 43] O( n 1 Kn 2 ) 3 L) is claimed to be the best previously known complexity bound for problem (10) This is incorrect. This entry should have been the bound, O( n 1 Kn 2 ) 0:5 n 2 2 (n 1 Kn 2 ) maxfn 1 ; n 2 g n 3 1 ] n 1 Kn 2 )L) obtained in Birge and Qi [3]. v) In [1] we derive cutting plane algorithms for two stage stochastic programs based on ellipsoids, volumetric centers and analytic centers. We also prove complexity results for these algorithms. The algorithm given in [1] based on volumetric centers applied to problem (10) would have the ....

J. R. Birge and L. Qi, Computing block-angular Karmarkar projections with applications to stochastic programming, Management Science 34 (1988) 1472-- 1479.

....7 5 (3:16) and if D 2 and W have full row rank we can define A s = 2 6 6 6 6 6 6 6 4 D 2 W W . W 3 7 7 7 7 7 7 7 5 : 3:17) The financial planning problems analysed in [16] have such a property. Another technique for correcting the rank of A s Theta Gamma1 s A T s was suggested in [4]. It is based on the transformation AA T = 2 6 6 6 6 4 DD T DS T 1 Delta Delta Delta DS T L S 1 D T S 1 S T 1 Delta Delta Delta S 1 S T L . SLD T SLS T 1 Delta Delta Delta SLS T L 3 7 7 7 7 5 2 6 6 6 6 4 WW T . WW T 3 7 7 7 7 5 = 2 6 6 6 6 ....

J.R. Birge and Liquin Qi, "Computing block-angular Karmarkar projections with applications to stochastic programming," Management Science 34(1988), 14721479.

....operation in an IPM iteration consists in forming the product of the scaled constraint matrix by its transpose and in computing the Cholesky factors of the resulting symmetric matrix, dense columns produce fill in that ruins the approach. Treating dense columns by a Schur complement technique [8] would preserve sparsity; moreover, the approach parallelizes well [30] Stochastic linear programs are naturally very sparse, and have a known sparsity structure, but this structure contains coupling columns (those associated with the first stage variables x) For this reason, standard IPM codes ....

J. R. Birge and L. Qi, Computing block-angular Karmarkar projections with applications to stochastic programming, Management Science, 34 (1988), pp. 1472--1479.

....Hertog [28] leads to new approaches to large scale optimization in general, and stochastic programming in particular. The idea of exploiting the stucture of blockangular linear programs and stochastic linear programs with recourse in interior point methods were first expressed by Birge and Qi [7], Lustig, Mulvey and Carpenter [35] Birge and Holmes [8] and Choi and Goldfarb [9] For the extension to problems with a staircase structure with a potential for parallel computing see Loute and Vial [34] The synergy between the principles of decomposition, partitioning and cutting planes with ....

J. R. Birge and L. Qi, "Computing block--angular Karmarkar projections with applications to stochastic programming", Management Science 34 (1988) 1472--1479.

....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 0 ; x 0 i P p i=1 hc i ; x i i subject to T i x 0 W i x i = b i i = 1; 2; p; x i 0; i = 1; 2; p; 5) where c i ; x i 2 R n i ; i = 0; 1; p; b i 2 R m i ; i = 1; p, and all matrices T i ; i = 1; p, and W i ; i = 1; ....

J. R. Birge and L. Qi, Computing block-angular Karmarkar projections with applications to stochastic programming, Management Science, 34 (1988), pp. 1472--1479.

....methods, respectively. See for example the book [6] It should be noted that when the matrix B has a block diagonal structure as in (3) we can decompose the inner maximization problem in (1) or the inner minimization problem in (2) into smaller subproblems of the same type. Many studies ([1, 2, 12], etc. have been done on how we exploit special structures such as generalized upper bounds, block diagonal structures and staircase structures in interiorpoint methods. The conceptual algorithms given in Section 6 has a close relation with a compact inverse implementation (see Section 9 of [2] ....

J. R. Birge and L. Qi, "Computing block-angular Karmarkar projections with applications to stochastic programming," Management Science 34 (1988) 1472--1479.

.... [880, 879] Amelioration [828] Analiza [49] analogous [208] Analyse [502] Analysis [22, 23, 502, 96, 139, 483, 486, 512, 595, 683, 751, 752, 795, 861, 912, 918, 952, 953, 960, 212] Analytic [465, 710, 268, 381, 376, 378, 383, 375, 377, 552, 763, 765, 767, 769, 766, 768, 896, 941] angular [90, 129, 731]. Ansatze [156] Anstreicher [810] Anticipated [584, 585, 587, 586, 809, 813, 823, 949] apparatus [76, 409, 876] Application [141, 422, 767, 57, 112, 232, 286, 454, 654, 673, 752, 759] Applications [766, 768, 84, 90, 110, 129, 359, 425, 481, 764, 790] Applied [782, 13, 190, 381, 783] ....

.... 378, 383, 375, 377, 552, 763, 765, 767, 769, 766, 768, 896, 941] angular [90, 129, 731] Ansatze [156] Anstreicher [810] Anticipated [584, 585, 587, 586, 809, 813, 823, 949] apparatus [76, 409, 876] Application [141, 422, 767, 57, 112, 232, 286, 454, 654, 673, 752, 759] Applications [766, 768, 84, 90, 110, 129, 359, 425, 481, 764, 790]. Applied [782, 13, 190, 381, 783] appraisal [168] approach [173, 187, 201, 202, 246, 311, 320, 360, 410, 419, 408, 438, 437, 482, 608, 680, 686, 685, 834, 895, 907, 931] Approaches [523, 254, 324, 407, 473, 522, 964] approchee [654, 787] Approximate [474, 654, 657, 894, 25, 57, 659, 709, ....

[Article contains additional citation context not shown here]

J. R. Birge and L. Qi. Computing block--angular Karmarkar projections with applications to stochastic programming. Management Science, 34:1472--1479, 1988.

....All efficient general purpose interior point codes [14] compute these orthogonal projections through a direct approach (Cholesky or Bunch Parlet) symmetric factorization [7] There exist structure exploiting specializations of these direct approaches for stochastic optimization. The method of [4, 5] handles the whole block of the first stage columns with the Schur complement mechanism. The approach proved to be efficient for problems in which the number of first stage variables, n 1 is not excessive. Additionally, as shown in [16] it parallelises in a scalable way. The method of [6] applies ....

....1 Delta Delta Delta T 1 T T S . T S T T 0 Delta Delta Delta T S T T S W S W T S 1 C C C C A : 25) This has an immediate negative influence on the Cholesky factor of AA T that becomes almost completely dense regardless of the sparsity of A. To avoid this degrading influence [4, 5] suggest special treatment of the whole block of the first stage columns T : it can be seen as a symmetric rank n 1 (n 1 is the dimension of x 1 ) corrector to a diagonal (hence very sparse) adjacency structure of the second stage columns. In this approach the first stage columns are taken into ....

Birge J. and L. Qi, (1988). Computing block-angular Karmarkar projections with applications to stochastic programming, Management Science 34, No 12, pp. 1472-1479.

....is the same as that for the conceptual algorithm in [9] There are other ways which can also exploit the special structure of programs without explicitly using the decomposition techniques. In the area of interior point methods, a technique called compact inverse implementation, for instance cf. [1] [2] 15] has been used. For certain problems, the interior point methods with decomposition (IPMwD) and the interior point methods with compact inverse implementation (IPMwCI) look similar. There are, however, substantial differences between IPMwD and IPMwCI: i) The IPMwCI treats all variables ....

Birge, J.R., L. Qi (1988). Computing block-angular Karmarkar projections with applications to stochastic programming. Management Science 34 1472--1479.

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J. R. Birge, L. Qi, Computing block-angular Karmarkar projections with applications to stachastic programming, Management Sci. 34, 1988, 14721479.

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Birge, J.R., Qi, L.: Computing block-angular Karmarkar projections with applications to stochastic programming, Management Sci., 34,1988, pp.1472-1479.

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J.R. Birge and L. Qi, 1988. Computing block-angular Karmarkar projections with applications to stochastic programming, Management Science 34, 1472-- 1479.

....solution of linear systems of the form ) dy = b: 1) Solving of this problem requires more then 90 95 of total programming time [1] Birge and Holmes [2] compared di erent methods for the solution of this system. They found that the factorization technique based on the work of Birge and Qi (BQ) [3] is more ecient and stable than other methods. They also suggested BQ for parallel computation. A parallel version of BQ for two stage stochastic programs was implemented on an Intel iPSC=860 hypercube and a Connection Machine CM 5 with nearly perfect speedup [4] According to our knowledge, this ....

....spent in communication and also by the ratio m i =n i . 8 4 Analysis of the BQ matrix factorization The matrix factorization ADA starts with a convenient expression of this matrix as a sum of two nonsingular matrices. This property is inevitable as it has been seen. Therefore the authors of [3] have expressed the element on the position (1; 1) as 0 I m 0 I m 0 : 33) But generally, one can use instead of I m 0 an arbitrary regular matrix or product of matrices XY. Naturally, the choice of such matrices has in uence on matrix G and on the computation of the system Gq = V p, too. ....

Birge, J.R., Qi, L.: Computing block-angular Karmarkar projections with applications to stachastic programming, Management Sci., 34,1988, pp.1472-1479.

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J.R. Birge and L. Qi. Computing block-angular Karmarkar projections with applications to stochastic programming. Management Science, 34(12):14721479, 1990.

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J. R. Birge and L. Qi, Computing block-angular Karmarkar projections with applications to stochastic programming, Management Sci. 34, 1988, pp. 1472-1479.

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J.R. Birge and L. Qi. Computing block-angular Karmarkar projections with applications to stochastic programming. Management Science, 34(12):14721479, 1990.

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J. R. Birge and L. Qi, "Computing block-angular Karmarkar projections with applications to stochastic programming," Management Science 34 (1988) 1472--1479.

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Birge, J.R., and Qi, L., Computing Block-Angular Karmarkar Projections with Applications to Stochastic Programming, Management Science, Vol. 34, pp. 1472--1479, 1988.

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J. R. Birge and L. Qi. Computing block-angular Karmarkar projections with applications to stochastic programming. Management Science, 34:1472--1479, 1988. Jeff Linderoth, Stephen Wright: Stochastic Programming on a Computational Grid

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J.R. Birge and L. Qi, Computing block-angular Karmarkar projections with applications to stochastic programming, Management Science, 34(1988), 1472-1479.

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Birge J.R. and L. Qi (1988), Computing Block-angular Karmarkar Projections with Applications to Stochastic Programming, Management Science, vol. 34, 1472-1479.

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Birge J.R. and L. Qi (1988), Computing Block-angular Karmarkar Projections with Applications to Stochastic Programming, Management Science, vol. 34, 1472-1479.

No context found.

J.R. Birge and L. Qi. Computing block-angular Karmarkar projections with applications to stochastic programming. Management Science, 34:1472--1479, 1988.

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Birge, J.R. and L. Qi (1988), Computing Block-Angular Karmarkar Projections with Applications to Stochastic Programming, Management Science, vol. 34, 1472-1479. Scenario Generation and Stochastic Programming Models for Asset Liability Management 23

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Birge J.R. and L. Qi (1988), Computing Block-angular Karmarkar Projections with Applications to Stochastic Programming, Management Science, vol. 34, 1472-1479.

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Birge J.R. and L. Qi (1988), Computing Block-angular Karmarkar Projections with Applications to Stochastic Programming, Management Science, vol. 34, 1472-1479.

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