Constructing test problems for bilevel evolutionary multi-objective optimization

Many real-world problems demand a feasible solution to satisfy physical equilibrium, stability, or certain properties which require an additional lower level optimization problem to be solved. Although such bilevel problems are studied somewhat in the context of a single objective in each level, there are not many studies in which multiple conflicting objectives are considered in each level. Bilevel multi-objective optimization problems offer additional complexities, as not every lower level Pareto-optimal front has a representative solution to the upper level Pareto-optimal front and that only a tiny fraction of participating lower level fronts make it to the upper level front. A couple of recent studies by the authors have suggested a viable EMO method to handle such problems. In this paper, we analyze the difficulties which a bilevel EMO procedure may face in handling such problems and present a systematic construction procedure for bilevel optimization test problems. Based on the suggested principles, we propose five test problems which are scalable in terms of number of variables and objectives, and which enable researchers to evaluate different phases of a bilevel problem solving task. The test problem construction procedure is interesting and may motivate other researchers to extend the idea to develop further test problems.