Easily Solvable Cases of Robust Discrete Optimization Problems

While the last chapter points out the difficulty associated with solving many classes of robust discrete optimization problems, the current chapter has an optimistic tone by describing polynomially solvable problems. The main source of difficulty of robust optimization problems comes from its min-max (or max-min) nature and its added dimensionality — the scenario sets. In many cases where both the decision variables and the scenario sets are continuous, the minimization and the maximization operations commute (order interchangeable), and thus the problem is much easier to solve. However, commutability requirement is a luxury in discrete optimization. Even the primal and its relaxation dual will in most cases inevitably lead to a gap between the corresponding objective values. We believe that the number of polynomially solvable discrete robust optimization problems is very limited.