In this paper we introduce four scenario Cluster based Lagrangian Decomposition procedures for obtaining strong lower bounds to the (optimal) solution value of two-stagestochastic mixed 0-1 problems. At each iteratio...
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In this paper we introduce four scenario Cluster based Lagrangian Decomposition procedures for obtaining strong lower bounds to the (optimal) solution value of two-stagestochastic mixed 0-1 problems. At each iteration of the Lagrangian based procedures, the traditional aim consists of obtaining the solution value of the corresponding Lagrangian dual via solving scenario submodels once the nonanticipativity constraints have been dualized. Instead of considering a splitting variable representation over the set of scenarios, we propose to decompose the model into a set of scenario clusters. We compare the computational performance of the four Lagrange multiplier updating procedures, namely the Subgradient Method, the Volume Algorithm, the Progressive Hedging Algorithm and the Dynamic Constrained Cutting Plane scheme for different numbers of scenario clusters and different dimensions of the original problem. Our computational experience shows that the Cluster based Lagrangian Decomposition bound and its computational effort depend on the number of scenario clusters to consider. In any case, our results show that the Cluster based Lagrangian Decomposition procedures outperform the traditional Lagrangian Decomposition scheme for single scenarios both in the quality of the bounds and computational effort. All the procedures have been implemented in a C++ experimental code. A broad computational experience is reported on a test of randomly generated instances by using the MIP solvers COIN-OR (2010, [18]) and CPLEX (2009, [17]) for the auxiliary mixed 0-1 cluster submodels, this last solver within the open source engine COIN-OR. We also give computational evidence of the model tightening effect that the preprocessing techniques, cut generation and appending and parallel computing tools have in stochasticinteger optimization. Finally, we have observed that the plain use of both solvers does not provide the optimal solution of the instances included in the testbed with which
In this paper we study solution methods for solving the dual problem corresponding to the Lagrangian Decomposition of two-stagestochastic mixed 0-1 models. We represent the two-stagestochastic mixed 0-1 problem by a...
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In this paper we study solution methods for solving the dual problem corresponding to the Lagrangian Decomposition of two-stagestochastic mixed 0-1 models. We represent the two-stagestochastic mixed 0-1 problem by a splitting variable representation of the deterministic equivalent model, where 0-1 and continuous variables appear at any stage. Lagrangian Decomposition (LD) is proposed for satisfying both the integrality constraints for the 0-1 variables and the non-anticipativity constraints. We compare the performance of four iterative algorithms based on dual Lagrangian Decomposition schemes: the Subgradient Method, the Volume Algorithm, the Progressive Hedging Algorithm, and the Dynamic Constrained Cutting Plane scheme. We test the tightness of the LD bounds in a testbed of medium- and large-scale stochastic instances.
We present an algorithmic approach for solving large-scale two-stagestochastic problems having mixed 0-1 first stage variables. The constraints in the first stage of the deterministic equivalent model have 0-1 variab...
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We present an algorithmic approach for solving large-scale two-stagestochastic problems having mixed 0-1 first stage variables. The constraints in the first stage of the deterministic equivalent model have 0-1 variables and continuous variables, while the constraints in the second stage have only continuous. The approach uses the twin node family concept within the algorithmic framework, the so-called branch-and-fix coordination, in order to satisfy the nonanticipativity constraints. At the same time we consider a scenario cluster Benders;decomposition scheme for solving large-scale LP submodels given at each TNF integer set. Some computational results are presented to demonstrate the efficiency of the proposed approach. (C) 2008 Elsevier Ltd. All rights reserved.
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