Cluster Lagrangean decomposition in multistage stochastic optimization

A parallel Branch-and-Fix Coordination based matheuristic algorithm for solving large sized multistage stochastic mixed 0–1 problems

“…Notice that now, by construction, the NAC are satisfied, but likely some of the other constraints are not; (c) Update the (linear) penalization term for each scenario submodel by using a subgradient estimator type of the non-implementable solution (i.e., the difference of the scenario solution with respect to the current implementable one) for each node; (d) Append the penalization term to each scenario submodel plus its weighted square function; (e) Iterate until a stopping criterion is satisfied. We conjecture that the convergence of PHA could be speed up by building scenario cluster submodels, instead of single scenario ones, by using the break stage based scheme presented in [26]. See in [33] an extension of the algorithm for two-stage and multistage by using by using a scenario cluster (bundle is called there) approach.…”

“…See our approach [26] for obtaining strong lower bounds to the solution values of small to medium sized instances, where the scenarios are distributed in clusters (also so-called bundles). The distribution reduces in a systematic way the number of NAC (non-anticipativity constraints that equate the scenario cluster variables in the splitting formulation of the model).…”

“…The scheme is based on a so-called break stage, such that the NAC of the variables to be dualized belong to the nodes in the scenario tree whose related stages are up to the break one. See also [24] as a specialization of [26] for providing a Lagrangean heuristic solution for the multistage stochastic pure 0-1 version of the difficult deterministic (combinatorial) facility location -assignment problem. The quasi-optimality gap that has been obtained for large sized instances is very small.…”

Fuzzy C-means-based scenario bundling for stochastic service network design

“…In particular, the covering strategy with commodity demand as scenario features yields the best performance amongst all of the proposed strategies. Besides these authors' works, scenario bundling is being vigorously studied under different names in other contexts, such as the multistage stochastic programs [12] and the chance-constrained programs [13].…”

“…First, the fuzzy partition matrix is initialized by a random number generator. To satisfy constraint (12), each entry in the matrix is divided by the sum of the elements in the same column. Next, the bundle centers and the membership scores are sequentially updated and the objective function is calculated in each iteration.…”