# Multi-Objective BiLevel Optimization by Bayesian Optimization
## 6 Conclusions  
Multi-objective bilevel optimization problems constitute one of the hardest classes of optimization models known, as they inherit the computational complexity of the hierarchical structure and optimization. We propose a hybrid algorithm, based on batch Bayesian optimization to reduce the computational cost of the problem. Our contribution is three-fold in the literature of multi-objective bilevel optimization. First, we embed two specifically designed acquisition function optimizers for multi-objective problems based on hypervolume improvement and parallel evaluations during optimization to approximate the leader's Pareto-front solutions. The proposed algorithm is explained in Section 4.1. Second, we discussed bilevel multi-objective optimization under upper-level uncertainty and presented another hybrid algorithm based on batch Bayesian optimization with _noisy_ hypervolume improvement in Section 4.2. Moreover, we explore how batch size selection affects the optimization process for each of the proposed approaches. Third, we evaluate our proposed algorithm on six numerical and two real-world problems selected from the environmental economics literature, which is a decision-making problem between an authority and the followers, and compare our results with state-of-the-art algorithms. The results show that the proposed algorithm improvesefficiency by significantly reducing computational cost. The proposed algorithm performed very competitively in terms of necessary function evaluation and convergence considering hypervolume improvement and IGD values. Future WorkThe preferences of decision are important when we consider the hierarchical decision-making processes. For instance, the follower's decision from the Pareto-optimal front affects the leader's problem and final solution set. In future work, we shall explore the decision-making from the Pareto-optimal front and how preference learning might affect the process. It would also be interesting to apply the proposed algorithm to the problems in the automated machine learning literature. For example, integrating the upper-level problem as a neural architecture search with considering multiple neural network models and lower-level problems to search the optimum hyperparameters for the candidate neural networks. As we mentioned with regard to the high-dimensional search space problem in Section 5.1, it might be interesting to explore how the proposed algorithm can be improved by using a high-dimensional Gaussian process [54] during the upper-level optimization. Moreover, we will look to gain some information by analyzing the relationship between the acquisition selection and the whole bilevel optimization results to observe how acquisition function selection affects the performance for obtaining the Pareto-optimal front at each iteration. Conceptualization, V.D. ; methodology, V.D. ; software, V.D. ; validation, V.D. and S.P. ; formal analysis, V.D. ; investigation, V.D. ; resources, V.D. ; data curation, V.D. ; writing--original draft preparation, V.D. ; writing--review and editing, S.P. ; visualization, V.D. ; supervision, S.P. ; funding acquisition, S.P. All authors have read and agreed to the published version of the manuscript. This publication has emanated from research conducted with the financial support of Science Foundation Ireland under Grant number 12/RC/2289-P2 at Insight the SFI Research Centre for Data Analytics at UCC, which is co-funded under the European Regional Development Fund. For Open Access, the author has applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission Data Availability Statement The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding authors. The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.