Most of the current expensive multiobjective optimization (MOO) algorithms focus on identifying Pareto optimal solutions. However, in some applications such as multiobjective modular design, decision-makers often prefer a set of optimal solutions that share common components in the decision space, which may conflict with Pareto optimality. Existing expensive MOO algorithms are not specifically designed to address this preference. To bridge this gap, we propose modeling the component-sharing preference in MOO as a special bi-level multiobjective optimization problem. Specifically, the upper-level is a single-objective optimization problem that seeks the optimal shared variables, while the lower-level is a multiobjective optimization problem aimed at identifying trade-off solutions for given shared variable values. Moreover, the lower-level objective is expensive-to-evaluate and can only be evaluated for a limited number of times. To efficiently solve this problem, we introduce a data-efficient algorithm called Bayesian Bi-level Search (BBS). The effectiveness of BBS is validated through six new benchmark problems and a real-world application involving the planform shape design of Blended-Wing-Body underwater glider. The results show that our method effectively identifies solutions with shared components within limited computational budgets.

In multiobjective optimization, the R2 indicator is widely used for designing the indicator-based algorithms, and the Tchebycheff approach is commonly employed in the decomposition-based algorithms. Despite their wide use, the connection between these two different paradigms is still not well understood, particularly in the field of multiobjective efficient global optimization (MOEGO). Considering that expected improvement (EI) is a cornerstone in efficient global optimization (EGO), this article first studies the relationship between R2-based EI and Tchebycheff-based EI. Then, we introduce a many-to-few (M2F) decomposition framework, offering a new perspective for linking the R2-based method and the Tchebycheff decomposition approach. By incorporating M2F decomposition into MOEGO, a new algorithm called R2/D-EGO is proposed. At each iteration, R2/D-EGO utilizes the Tchebycheff decomposition paradigm to generate a set of candidate solutions, each one corresponding to a different weight vector. Subsequently, a subset of query points is selected from the candidates based on the lower bound of R2-based EI. Empirical results indicate that the proposed R2/D-EGO is highly competitive in comparison with both the R2-based and decomposition-based MOEGO algorithms in the parallel (or batch) setting.

Multiobjective optimization problems (MOPs) are prevalent in machine learning, with applications in multi-task learning, learning under fairness or robustness constraints, etc. Instead of reducing multiple objective functions into a scalar objective, MOPs aim to optimize for the so-called Pareto optimality or Pareto set learning, which involves optimizing more than one objective function simultaneously, over models with thousands to millions of parameters. Existing benchmark libraries for MOPs mainly focus on evolutionary algorithms, most of which are zeroth-order or meta-heuristic methods that do not effectively utilize higher-order information from objectives and cannot scale to large-scale models with millions of parameters. In light of the above challenges, this paper introduces \algoname, the first multiobjective optimization library that supports state-of-the-art gradient-based methods, provides a fair and comprehensive benchmark, and is open-sourced for the community.

The hypervolume metric is widely used to guide the search in multiobjective optimization. However, in parallel expensive multiobjective optimization, the hypervolume-based multipoint expected improvement (EI) suffers from high computational overhead and scales poorly with the batch size. To address this issue, we integrate hypervolume-based EI with the MOEA/D framework and propose a novel EI, named the expected direction-based hypervolume improvement (DirHV-EI). The DirHV-EI only measures the hypervolume improvement within each axis-parallel box induced by the modified Tchebycheff scalarization. Thus, it has a simple analytical expression that can be easily computed. Theoretical analysis indicates that the maximization of our proposed improvement function can help to maximize both the weighted hypervolume and the Tchebycheff improvement metrics. Using DirHV-EI, we design a decomposition-based Bayesian optimization algorithm for solving expensive multiobjective optimization problems. At each iteration, the MOEA/D is used to maximize the DirHV-EI values with respect to a number of direction vectors in a collaborative manner, and a number of candidate solutions can be obtained. Then, a submodularity-based greedy selection strategy is used to select multiple query points from the candidates. Experimental results on both benchmark instances and real-world problems show that our proposed algorithm is an efficient and effective method for parallel expensive multiobjective optimization.

Expected improvement (EI) is a widely used acqui-sition function in single-objective Bayesian optimization (BO). To deal with expensive multiobjective optimization problems, many extensions of EI have been proposed. Among them, expected Tchebycheff improvement (ETI) is a representative approach. Unfortunately, the calculation of exact ETI values remains a challenging task when leveraging Gaussian process models to model each objective. Therefore, approximation methods are commonly employed. However, the estimation error of these methods cannot be fully determined without the aid of exact formulas. Borrowing the idea of the exact formulas previously developed for the computation of multi-point EI in single-objective BO, this article gives an exact formula for the computation of ETI with any number of objectives. Numerical experiments are conducted to compare the performance of our exact formula and a commonly- used approximation method. Considering the similarity of ETI and multi-point EI, we also extend the approximation method designed for ETI to estimate the values of multi-point EI efficiently.