Multistage Robust Mixed-Integer Optimization with Adaptive Partitions

Abstract

We present a new method for multistage adaptive mixed integer optimization (AMIO) problems that extends previous work on finite adaptability. The approach analyzes the optimal solution to a non-adaptive version of an AMIO problem to gain insight into which regions of the uncertainty set are restricting the objective function value. We use this information to construct partitions in the uncertainty set, leading to a finitely adaptable formulation of the AMIO problem. We repeat this process iteratively to further improve the objective until appropriate termination criteria are reached. We provide theoretical motivation for this method, and fully detail how to apply finite adaptability to multistage AMIO problems in a way that respects the natural nonanticipativity constraints. We provide lower and upper bounds on the solution quality that have implications for computational tractability. Finally we demonstrate in computational experiments that the method can provide substantial improvements over a non-adaptive solution and existing methods. In particular, we find that our method delivers high-quality solutions even as the problem scales in both number of time stages and in the number of decision variables.

Solving the bifurcated and nonbifurcated robust network loading problem with k‐adaptive routing

“…As mentioned before, choosing the optimal partition makes the problem extremely difficult so that we focus in the heuristic solution algorithm proposed by [12].…”

“…where only one nominal element will be selected for each constraint involved in (12). If we were to remove all possible constraints that correspond to uncertain parameters that are not active at optimality, we are left with a reduced set of constraints that is sufficient to constrain the solution to its value.…”

“…The authors of [12] use the active uncertain parameters to construct a particular partition. Applied to kRN L, their approach imposes that each indexed element selectedd k , k ∈ {1, ..., |E|} ofD belongs to a single subset of the new partition, using Voronoi diagrams.…”

“…The first two were already suggested in [12], while the decomposition algorithm and valid inequalities are specific to kRN L studied herein. Nested partial partitioning In our problem, there are |E| capacity constraints leading to a total of as much of |E| active uncertain parameters and, so, |E| subsets associated to the repartition of each subset D k of the previous partition.…”

“…• We show the application of iterative k-adaptive partitioning algorithm defined in [12] to RN L. As a side effect we apply decomposition techniques where we verify performance of newly CPLEX 12.7 Benders Decomposition functions.…”