On The Recoverable Robust Traveling Salesman Problem

We consider an uncertain traveling salesman problem, where distances between nodes are not known exactly, but may stem from an uncertainty set of possible scenarios. This uncertainty set is given as intervals with an additional bound on the number of distances that may deviate from their expected, nominal value. A recoverable robust model is proposed, that allows a tour to change a bounded number of edges once a scenario becomes known. As the model contains an exponential number of constraints and variables, an iterative algorithm is proposed, in which tours and scenarios are computed alternately. While this approach is able to find a provably optimal solution to the robust model, it also needs to solve increasingly complex subproblems. Therefore, we also consider heuristic solution procedures based on local search moves using a heuristic estimate of the actual objective function. In computational experiments, these approaches are compared. Finally, an alternative recovery model is discussed, where a second-stage recovery tour is not required to visit all nodes of the graph. We show that the previously NP-hard evaluation of a fixed solution now becomes solvable in polynomial time

Algorithms and uncertainty sets for data-driven robust shortest path problems

We consider robust shortest path problems, where the aim is to find a path that optimizes the worst-case performance over an uncertainty set containing all relevant scenarios for arc costs. The usual approach for such problems is to assume this uncertainty set given by an expert who can advise on the shape and size of the set. Following the idea of data-driven robust optimization, we instead construct a range of uncertainty sets from the current literature based on real-world traffic measurements provided by the City of Chicago. We then compare the performance of the resulting robust paths within and outside the sample, which allows us to draw conclusions on the suitability of uncertainty sets. Based on our experiments, we then focus on ellipsoidal uncertainty sets, and develop a new solution algorithm that significantly outperforms a state-of-the art solver

On The Recoverable Robust Traveling Salesman Problem

We consider an uncertain traveling salesman problem, where distances between nodes are not known exactly, but may stem from an uncertainty set of possible scenarios. This uncertainty set is given as intervals with an additional bound on the number of distances that may deviate from their expected, nominal value. A recoverable robust model is proposed, that allows a tour to change a bounded number of edges once a scenario becomes known. As the model contains an exponential number of constraints and variables, an iterative algorithm is proposed, in which tours and scenarios are computed alternately. While this approach is able to find a provably optimal solution to the robust model, it also needs to solve increasingly complex subproblems. Therefore, we also consider heuristic solution procedures based on local search moves using a heuristic estimate of the actual objective function. In computational experiments, these approaches are compared. Finally, an alternative recovery model is discussed, where a second-stage recovery tour is not required to visit all nodes of the graph. We show that the previously NP-hard evaluation of a fixed solution now becomes solvable in polynomial time

Alternative Formulations for the Ordered Weighted Averaging Objective

The ordered weighted averaging objective (OWA) is an aggregate function over multiple optimization criteria which received increasing attention by the research community over the last decade. Different to the ordered weighted sum, weights are attached to ordered objective functions (i.e., a weight for the largest value, a weight for the second-largest value and so on). As this contains max-min or worst-case optimization as a special case, OWA can also be considered as an alternative approach to robust optimization. For linear programs with OWA objective, compact reformulations exist, which result in extended linear programs. We present new such reformulation models with reduced size. A computational comparison indicates that these formulations improve solution times