Evolutionary Bilevel Optimization Based on Covariance Matrix Adaptation

Bilevel optimization refers to a challenging optimization problem which contains two levels of optimization problems. The task of bilevel optimization is to find the optimum of the upper-level problem, subject to the optimality of the corresponding lower-level problem. This nested nature introduces many difficulties such as nonconvexity and disconnectedness, and poses great challenges to traditional optimization methods. Using evolutionary algorithms in bilevel optimization has been demonstrated to be very promising in recent years. However, these algorithms suffer from low efficiency since they usually require a huge number of function evaluations. This paper proposes a bilevel covariance matrix adaptation evolution strategy to handle bilevel optimization problems. A search distribution sharing mechanism is designed so that we can extract a priori knowledge of the lower-level problem from the upper-level optimizer, which significantly reduces the number of function evaluations. We also propose a refinement-based elite preservation mechanism to trace the elite and avoid inaccurate solutions. Comparisons with five state-of-the-art algorithms on 22 benchmark problems and two real-world applications are carried out to test the performance of the proposed approach. The experimental results have shown the effectiveness of the proposed approach in keeping a good tradeoff between solution quality and computational efficiency.