A Knee Point Driven Evolutionary Algorithm for Multiobjective Bilevel Optimization

Bilevel optimization is a special type of optimization in which one problem is embedded within another. The bilevel optimization problem (BLOP) of which both levels are multiobjective functions is usually called the multiobjective BLOP (MBLOP). The expensive computation and nested features make it challenging to solve. Most existing studies look for complete lower-level solutions for every upper-level variable. However, not every lower-level solution will participate in the bilevel Pareto-optimal front. Under a limited computational budget, instead of wasting resources to find complete lower-level solutions that may not be in the feasible region or inducible region of the MBLOP, it is better to concentrate on finding the solutions with better performance. Bearing these considerations in mind, we propose a multiobjective bilevel optimization solving routine combined with a knee point driven algorithm. Specifically, the proposed algorithm aims to quickly find feasible solutions considering the lower-level constraints in the first stage and then concentrates the computational resources on finding solutions with better performance. Besides, we develop several multiobjective bilevel test problems with different properties, such as scalable, deceptive, convexity, and (dis)continuous. Finally, the performance of the algorithm is validated on a practical petroleum refining bilevel problem, which involves a multiobjective environmental regulation problem and a petroleum refining operational problem. Comprehensive experiments fully demonstrate the effectiveness of our presented algorithm in solving MBLOPs.