Decomposition approaches for recoverable robust optimization problems

• We present two decomposition approaches for recoverable robust optimization.
• We prove that the LP-relaxation of one of the two models is stronger than the other.
• We present a column generation algorithm for the size-robust knapsack problem.
• We present a column generation algorithm for the demand robust shortest path problem.
• Our approach is very promising and can be generalized to many other problems.

Real-life planning problems are often complicated by the occurrence of disturbances, which imply that the original plan cannot be followed anymore and some recovery action must be taken to cope with the disturbance. In such a situation it is worthwhile to arm yourself against possible disturbances by including recourse actions in your planning strategy. Well-known approaches to create plans that take possible, common disturbances into account are robust optimization and stochastic programming. More recently, another approach has been developed that combines the best of these two: recoverable robustness. In this paper, we solve recoverable robust optimization problems by the technique of branch-and-price. We consider two types of decomposition approaches: separate recovery and combined recovery. We will show that with respect to the value of the LP-relaxation combined recovery dominates separate recovery. We investigate our approach for two example problems: the size robust knapsack problem, in which the knapsack size may get reduced, and the demand robust shortest path problem, in which the sink is uncertain and the cost of edges may increase. For each problem, we present elaborate computational experiments. We think that our approach is very promising and can be generalized to many other problems.