Robust combinatorial optimization under budgeted-ellipsoidal uncertainty

In the field of robust optimization, uncertain data are modeled by uncertainty sets which contain all relevant outcomes of the uncertain problem parameters. The complexity of the related robust problem depends strongly on the shape of the chosen set. Two popular classes of uncertainty are budgeted uncertainty and ellipsoidal uncertainty. In this paper, we introduce a new uncertainty class which is a combination of both. More precisely, we consider ellipsoidal uncertainty sets with the additional restriction that at most a certain number of ellipsoid axes can be used at the same time to describe a scenario. We define a discrete and a convex variant of the latter set and prove that in both cases the corresponding robust min-max problem is NP-hard for several combinatorial problems. Furthermore, we prove that for uncorrelated budgeted-ellipsoidal uncertainty in both cases the min-max problem can be solved in polynomial time for several combinatorial problems if the number of axes which can be used at the same time is fixed. We derive exact solution methods and formulations for the problem which we test on random instances of the knapsack problem and of the shortest path problem.