Faster Algorithms for Min-max-min Robustness for Combinatorial Problems with Budgeted Uncertainty

International audienceWe consider robust combinatorial optimization problems where the decision maker can react to a scenario by choosing from a finite set of k solutions. This approach is appropriate for decision problems under uncertainty where the implementation of decisions requires preparing the ground. We focus on the case that the set of possible scenarios is described through a budgeted uncertainty set and provide three algorithms for the problem. The first algorithm solves heuristically the dualized problem, a non-convex mixed-integer non-linear program (MINLP), via an alternating optimization approach. The second algorithm solves the MINLP exactly for k = 2 through a dedicated spatial branch-and-bound algorithm. The third approach enumerates k-tuples, relying on strong bounds to avoid a complete enumeration. We test our methods on shortest path instances that were used in the previous literature and on randomly generated knapsack instances, and find that our methods considerably outperform previous approaches. Many instances that were previously not solved within hours can now be solved within few minutes, often even faster

The resource constrained shortest path problem with uncertain data: a robust formulation and optimal solution approach

International audienceThe Resource Constrained Shortest Path Problem (RCSP P) models several applications in the fields of transportation and communications. The classical problem supposes that the resource consumptions and the costs are certain and looks for the cheapest feasible path. These parameters are however hardly known with precision in real applications, so that the deterministic solution is likely to be infeasible or suboptimal. We address this issue by considering a robust counterpart of the RCSP P. We focus here on resource variation and model its variability through the uncertainty set defined by Bertismas and Sim (2003,2004), which can model the risk aversion of the decision maker through a budget of uncertainty. We solve the resulting problem to optimality through the well-known three phase approach dealing with bounds computation, network reduction and gap closing. In particular, we compute robust bounds on the resource consumption and cost by solving the robust shortest path problem and the dual robust Lagrangian relaxation, respectively. Dynamic programming is used to close the duality gap. Upper and lower bounds are used to reduce the dimension of the network and incorporated in the dynamic programming in order to fathom unpromising states. An extensive computational phase is carried out in order to asses the behavior of the defined strategy comparing its performance with the state-of-the-art. The results highlight the effectiveness of our approach in solving to optimality * 1 benchmark instances for RCSP P when Γ is not too large, tailored for the robust counterpart. For larger values of Γ, we show that the most efficient method combines deterministic preprocesing with the iterative algorithm from Bertsimas and Sim (2003). We also illustrate the failure probability of the robust solutions through Monte Carlo sampling

A note on the Bertsimas & Sim algorithm for robust combinatorial optimization problems

We improve the well-known result presented in Bertsimas and Sim (Math Program B98:49-71, 2003) regarding the computation of optimal solutions of Robust Combinatorial Optimization problems with interval uncertainty in the objective function coefficients. We also extend this improvement to a more general class of Combinatorial Optimization problems with interval uncertainty