The bilevel optimisation problem is an optimisation problem, representing the upper-level and leader, which has as its constraints another optimisation problem, representing the lower-level and follower. Several challenges and complexities appear in order to reach the optimality of this problem. When no assumptions about the problem are made a solution is hard to find, leading the community to approach it both with classical and evolutionary methods.
Worst-case scenario optimisation – also known as min-max optimisation – is a special instance of the bilevel problem. It deals with the minimisation of the maximum output in all scenarios of a given problem. One can find these problems in optimisation under uncertainty, where the lower level plays the role of nature, that reacts to upper level’s decisions in the most catastrophic way. Therefore, the upper level aims to find a solution that performs best even in the worst possible lower level’s reaction. The most conservative but also the most robust solution is obtained by this approach.
There exist various Bilevel Evolutionary Algorithms (BLEAs), that have been tested and performed well on bilevel optimisation problems. These algorithms should also perform well in solving the worst-case scenario optimisation problem. Indeed in our previous research , we tested 3 BLEAs on 13 min-max synthetic problems, where they reached the near-optimal solutions in most of the cases. The BLEAs tested are BLDE,
BLEAQ  and BL-CMA-ES.
Extending this research, we apply them to a real-world engineering optimisation problem under uncertainty. The aforementioned problem is a simple truss design problem subjected to loads with uncertainty in magnitude and/or direction. The algorithms aim to find the optimal design under the worst-case scenario.