Parallel Decomposition Procedures for Large-scale Linear Programming Problems

In practice, many large-scale linear programming problems are too large to be solved effectively due to the computer\u27s speed and/or memory limitation, even though today\u27s computers have many more capabilities than before. Algorithms are exploited to solve such large linear programming problems, either in the sequential or parallel computation environment. This study focuses on two parallel algorithms for solving large-scale linear programming problems efficiently. The first parallel decomposition algorithm discussed in this study is from the theory problems in a special block-angular structure. The theory or the decomposition principle is first examined. Since the subproblems of a linear programming problem can be in any of the three possible cases—optimal solution case, unbounded solution case and no solution case, examples are provided for solving the problem when its subproblems are in any of these cases. The concept of extreme directions is discussed due to its direct connection with the unbounded solution case. A parallel computation code, which can handle all these cases, is implemented in this study with the decomposition principle theory and its performance is tested for large-scale linear programming problems. Only the problems in the special block-angular structure can be solved with the decomposition principle. For general linear programming problems, this study proposed a new decomposition algorithm named “division by the interior point”. The idea of this new algorithm is as follows: with a found interior point inside the feasible region, divide the feasible region into multiple subregions and use multiple processors to solve the problem in each subregion. This new algorithm is first demonstrated with a few small numerical examples. A parallel computation code in this new idea is implemented and tested with large-scale linear programming problems

A Two-Phase Decomposition Algorithm for Linear Programs with Angular Structure

Large scale linear programming problems often have special forms of constraints. An angular structure is a typical instance. The Dantzig-Wolfe decomposition principle is an effective tool for solving the linear programming problem with angular structure. So far, the decomposition principle has been used only in the second-phase problem of the two-phase simplex procedure. This paper proposes a complete two-phase algorithm, in which the decomposition technique is fully utilized both in the first and the second phases. The present algorithm is then applicable, without any a priori knowledge of an initial feasible solution, to all the classes of linear programs with angular structure, though it may have some computational redundancies

Regularized Decomposition of Stochastic Programs: Algorithmic Techniques and Numerical Results

A finitely convergent non-simplex method for large scale structured linear programming problems arising in stochastic programming is presented. The method combines the ideas of the Dantzig-Wolfe decomposition principle and modern nonsmooth optimization methods. Algorithmic techniques taking advantage of properties of stochastic programs are described and numerical results for large real world problems reported

On the Regularized Decomposition Method for Two Stage Stochastic Linear Problems

A new approach to the regularized decomposition (RD) algorithm for two stage stochastic problems is presented. The RD method combines the ideas of the Dantzig-Wolfe decomposition principle and modern nonsmooth optimization methods. A new subproblem solution method using the primal simplex algorithm for linear programming is proposed and then tested on a number of large scale problems. The new approach makes it possible to use a more general problem formulation and thus allows considerably more freedom when creating the model. The computational results are highly encouraging

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Recently, the remarkable developments of workstations, such as the high perfomance and the mark down of the costs, have rapidly popularized the use of workstations. Also, the problem solving style constituted by most of the workstations is well utilitized. By using the network communication ability, distributed processing can be considered. It is possible to have systems that take much less processing time by using distributed processing than concentrated processing. Moreover, many algorithms have been proposed to solve the problems of large scale linear programming. Usually, the solving of large scale linear programming takes a tremendous amount of time. However, this kind of problem usually consists of many zero coefficient variables. Thus, the original problem breaks down into many smaller, independent problems. The original problem become to be structuralized. This kind of problem can be well solved by using Dantzig-Wolfe\u27s decomposition principle. Decomposition principle takes the original problem and divides it one into central problem and many sub-problems. Therefore, distributed Processing is a good application of the decomposition principle. This reseach develops the system that solves the structuralized problem, and the algorithm of solving large scale linear programming problems is also discussed

Mechanism Design via Dantzig-Wolfe Decomposition

In random allocation rules, typically first an optimal fractional point is calculated via solving a linear program. The calculated point represents a fractional assignment of objects or more generally packages of objects to agents. In order to implement an expected assignment, the mechanism designer must decompose the fractional point into integer solutions, each satisfying underlying constraints. The resulting convex combination can then be viewed as a probability distribution over feasible assignments out of which a random assignment can be sampled. This approach has been successfully employed in combinatorial optimization as well as mechanism design with or without money. In this paper, we show that both finding the optimal fractional point as well as its decomposition into integer solutions can be done at once. We propose an appropriate linear program which provides the desired solution. We show that the linear program can be solved via Dantzig-Wolfe decomposition. Dantzig-Wolfe decomposition is a direct implementation of the revised simplex method which is well known to be highly efficient in practice. We also show how to use the Benders decomposition as an alternative method to solve the problem. The proposed method can also find a decomposition into integer solutions when the fractional point is readily present perhaps as an outcome of other algorithms rather than linear programming. The resulting convex decomposition in this case is tight in terms of the number of integer points according to the Carath{\'e}odory's theorem