Effective Identification of Lower-Level Optimal Solutions via Discriminator of Conditional Generative Adversarial Network

Bilevel multi-objective optimization problem (BLMOP) can be seen as a special constrained multi-objective optimization problem (CMOP), where the optimality constraint of the lower-level (LL) problem cannot be easily and quickly checked, but must be verified by solving the LL problem. If there was a relatively simple alternative formulation for effectively identifying LL optimal solutions, the abovementioned special constraint could become as simple as the usual constraints and the nested optimization structure of BLMOP would be altered, which can greatly improve efficiency and effectively find LL optimal solutions with excellent upper-level (UL) performance. In this paper, we improve the training mechanism for conditional generative adversarial network (cGAN) by introducing multiple generators to construct a reasonable reference distribution, so as to prevent the discriminator from performance degradation. With a discriminator identifying LL optimal solutions effectively, the nested optimization structure of BLMOP is altered by adding the objective that maximizing the discriminator output score into the UL objectives and removing the LL optimality constraint, realizing synchronous optimization of UL and LL vectors. The cooperation of generators and discriminator greatly reduces the computational overhead for solving BLMOPs. The proposed algorithm has achieved the best or competitive results in comparison with 6 state-of-the-art algorithms and a nested method on benchmark problems and a real-world problem, whose effectiveness has been demonstrated.