# Multi-Objective BiLevel Optimization by Bayesian Optimization
## 1 Introduction
The benefit of this approach is that it makes parallel evaluations possible. Parallel evaluations are a vital part of various kinds of practical applications such as product design in the food industry, which can be modelled as bilevel problems with multiple objectives [18]. One can only produce a relatively small batch of different products at one time. Product quality, especially for foods, is usually highly dependent on the processing time since production. It is usually best to evaluate a batch immediately. So the next batch of products should be designed based on the feedback so far. Many approaches have been developed for solving multi-objective bilevel problems. A particle swarm optimization algorithm is used to compute optimistic and pessimistic solutions in [19], who also developed a differential evolution algorithm to compute the four extreme solutions in [20]. Ref. [21] develop a nested differential evolution-based algorithm for multi-objective bilevel problems (DBMA). However, the necessary number of function evaluations is high, due to the nature of evolutionary algorithms. Satisfactory solutions are computed using leader preferences in [22]. Ref. [23] propose a hybrid evolutionary algorithm (H-BLEMO) but again gigantic computational cost was a problem. Ref. [24] develop a hybrid particle swarm optimization algorithm. Ref. [25] use a multi-objective particle swarm optimization algorithm and efficiently computes lower-level solutions (OMOPSO-BL). More details can be found in [20] on optimistic multi-objective bilevel problems. Much work has been performed in the Bayesian literature on batch selection, and many developed specialized acquisition functions for batch Bayesian optimization. Some, such as qEI [26], qKG [27] and local penalization [28] methods, search for optimal batch selection. We use tailored acquisition functions to optimize the design space for multi-objective problems with fewer expensive function calls. To the best of our knowledge, there is no previous work on a batch Bayesian optimization approach to multi-objective bilevel problems [29]. The closest work is that of [30], which uses multiple Gaussian models for multiple objectives to solve robot and behaviour co-design, but not with batch selection during Bayesian optimization with upper-level objective uncertainty or with hypervolume improvement. The aim of this paper is to improve computational efficiency to solve multi-objective bilevel problems while using hypervolume improvement to approximate the leader'sPareto-optimal solution set. We compare the proposed algorithm with some existing algorithms in terms of function evaluations and hypervolume improvement. Moreover, we show how batch selection affects performance in batch Bayesian optimization. We propose two hybrid algorithms with a multi-objective Bayesian optimization approach to solve noise-free and noisy multi-objective bilevel problems, using two different acquisition optimizers specifically designed for multi-objective optimization. We use a non-dominated sorting genetic algorithm (NSGA-II) [31] to solve the lower-level multi-objective problems, with a crossover operator to decrease lower-level function evaluations. Using Bayesian batch optimization via a multi-output GP surrogate model in a multi-objective bilevel setting allows us to measure and handle uncertainties in the multi-objective upper-level design space. Selecting multiple points rather than one point at a time makes the optimization process more efficient. Moreover, the multi-output GP allows us to work with several kernels that worked well in the literature while optimizing the multi-objective upper-level design space. The contribution of this work is threefold. First, we present two hybrid algorithms for solving multi-objective bilevel problems with noise-free settings, which is an extension of the workshop paper [32], and a noisy setting is added using Bayesian batch optimization. Second, batch Bayesian optimization allows us to use batch selection during optimization rather than just once per iteration.