Cluster Lagrangean decomposition in multistage stochastic optimization

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).