Subtree decomposition for multistage stochastic programs

A number of methods for solving multistage stochastic linear programs with recourse decompose the deterministic equivalent [4] to form subproblems based on scenarios (e.g., Rockafellar and Wets [6] and Mulvey and Ruszczyński [5]). Other methods use forms of Benders decomposition to form subproblems based on nodes of the scenario tree (e.g., Birge [1] and Gassmann [2]). Both types of algorithms can be conceptualized as a decomposition of the scenario tree [4] of the stochastic program. Also, under both of these approaches the number and size of the subproblems is predefined: each scenario or node corresponds to a subproblem. This research describes an algorithmic approach which allows more flexibility in the structure and size of the subproblems. Two existing decomposition approaches are used as a basis for applying the principles. The deterministic equivalent is reformulated in two ways in order to meet the needs of two underlying decomposition approaches. We use the concept of the extension of a node set, which adds the predecessor of each node to the node set. A subtree cover of the scenario tree is a set covering of the scenario nodes with the property that for each node set the subgraph induced by its extension is connected. A subtree partition of the scenario tree is a partition of the scenario nodes with the property that the subgraph induced by each node set is connected. Note the different node sets that are required to induce subtrees for the above. Different subtree covers and subtree partitions will lead to different subtree decompositions for the same problem.