Cluster Lagrangean decomposition in multistage stochastic optimization

• A Multistage scenario Cluster Lagrangian Decomposition (MCLD) approach for obtaining strong lower bounds on the solution value of large sized instances of the multistage stochastic mixed 0–1 problem is presented.
• MCLD frequently gives the solution value of the original stochastic mixed 0–1 problem by using merged or pure scenario cluster submodels either without Lagrange multipliers usage or considering any of the four Lagrange multipliers schemes that we have selected to work with.
• It is easy to prove that the performance of the MCLD procedure outperforms the traditional Lagrangian Decomposition scheme based on single scenarios in both the bound׳s quality and elapsed time.
• The efficiency of the four updating schemes, as contrasted in the testbed we have experimented with, is very similar in quality of the MCLD bound. However, a careful implementation of the Subgradient Method makes it more robust than the other three updating schemes. Notice that the SM based MCLD approach obtains the solution value in the original problem in thirteen out of the fourteen instances included in the testbed.
• The proposed approach requires an elapsed time that is one order of magnitude smaller than the time required by CPLEX, in the worst case.

We present a Lagrangean Decomposition approach for obtaining strong lower bounds on minimizing medium to large scale multistage stochastic mixed 0–1 problems. The problem is represented by a mixture of the splitting representation up to a given stage, so-named break stage, and the compact representation for the other stages along the time horizon. The dualization of the nonanticipativity constraints for the variables up to the break stage results in a model that can be decomposed into a set of independent scenario cluster submodels. The nonanticipativity constraints for the 0-1 and continuous variables in the cluster submodels are implicitly satisfied. Four scenario cluster schemes are compared for Lagrangean multipliers updating such as the Subgradient Method, the Volume Algorithm, the Lagrangean Progressive Hedging Algorithm and the Dynamic Constrained Cutting Plane scheme. We have observed in randomly generated instances that the smaller the number of clusters, the stronger the lower bound provided for the original problem (even, frequently, it is the solution value).