Combinatorial Optimization with Noisy Inputs: How Can We Separate the Wheat from the Chaff?
We postulate that real world data are almost always noisy, and an exact solution to a noisy input instance of a combinatorial optimization problem is not what we really want. Noise, or input data uncertainty, has a variety of reasons, such as for instance the need to estimate data based on imprecise measurements or on predictions (drawn from historical data and expected modifications). There is a variety of popular ways to deal with this uncertainty problem. In lucky cases in which the input data distribution is known, one might aim at obtaining a solution that is good in expectation. A different, promising way to handle uncertainty is based on the availability of a discrete set of possible problem instances (sometimes reflecting a distribution), so-called scenarios. A solution must be proposed for a set of scenarios as input, and thereafter a single scenario reveals itself as the actual one. The goal here is to achieve a high quality of the proposed solution with respect to the revealed scenario. Stochastic programming can be used to aim at a good solution in expectation that is feasible for most scenarios. In contrast, robust optimization most often aims at a solution that is feasible in all scenarios and has smallest worst case cost. In any case, uncertainty is considered a curse, a burden, a difficult problem that needs to be dealt with at extra computational cost.