Parallel bundle-based decomposition for large-scale structured mathematical programming problems

On the efficacy of distributed simplex algorithms for linear programming

“…Medhi [ 16] developed a bundle-based distributed decomposition algorithm for convex optimization problems, and demonstrated its effectiveness on block-angular linear programs as a special class of convex programs. Likewise, Ho et al [11] implemented Dantzig-Wolfe decomposition [3] on the CRYSTAL multicomputer [4], and showed that multiprocessing can be potentially useful for solving large-scale structured linear programs.…”

“…The number of such independent blocks, that could be comfortably addressed, is increasing due to the incorporation of more and more processors into such computers. While the asynchronous, multiprocessing capability of MIMD computers has been successfully exploited to solve certain classes of structured linear programs (LP's) [9,11,16], its effectiveness with unstructured LP's has not been fully studied. In this paper, we take up this topic by considering the adaptation of MIMD computers to solve general linear programs in the form:…”

Symmetric and Asymmetric Parallelization of a Cost-Decomposition Algorithm for Multicommodity Flow Problems

“…the relaxed constraints. Lagrangean-based approaches such as the Dantzig-Wolfe method (Ho et al 1988, Gnanendran andHo 1993) or the bundle method (Medhi 1990, Ferris and Horn 1998, Frangioni and Gallo 1999 directly approach the maximization of the nondifferentiable Lagrangean function, whose calculation breaks down in the solution of an independent subproblem for each block. Other methods are based on differentiable but non-separable functions, such as the augmented Lagrangean (De Leone et al 1994, Kontogiorgis et al 1996, linear-quadratic (Pinar and Zenios 1992) or exponential (Grigoriadis and Khachiyan 1995) penalty functions, or logarithmic barrier functions Schultz 1992, De Leone et al 1994).…”

“…Approaches based on complex coordinators are the Dantzig-Wolfe method (Ho et al 1988, Gnanendran andHo 1993) and bundle methods (Medhi 1990, De Leone et al 1993, Ferris and Horn 1998, Frangioni and Gallo 1999. At each step, the master uses (potentially all) the information collected from all the previous slave phases-columns or subgradients, depending on the viewpoint-to compute the new vector of prices to be broadcasted to the slaves.…”

“…In the last few years, many parallel approaches to MMCF have been developed (Zenios and Mulvey 1988;Medhi 1990;Zenios 1991;Mangasarian 1992, 1994;Lustig and Li 1992;Pinar and Zenios 1992;Gnanendran and Ho 1993;De Leone et al 1993Zenios 1993, Kontogiorgis et al 1996De Silva and Abramson 1998;Ferris and Horn 1998;Castro and Frangioni 2001), demonstrating the broad interest in the solution of large-scale MMCF problems. The present approach can be classified among those with complex coordinator, that seem to be best suited for implementation on coarse-grained massively parallel machines.…”