Multi-Objective Bayesian Optimization for Networked Black-Box Systems: A Path to Greener Profits and Smarter Designs

TL;DR

MOBONS introduces a unifying framework for multi-objective optimization of networked, potentially cyclic, grey-box systems by modeling each network node with Gaussian processes and employing Thompson sampling to efficiently explore the design space. By explicitly incorporating the network structure through a function network $\boldsymbol{Y} = \boldsymbol{F}(\boldsymbol{x}, \boldsymbol{Y})$ and a projection $\boldsymbol{G}(\boldsymbol{x}) = \boldsymbol{C} \boldsymbol{Y}^*(\boldsymbol{x})$, MOBONS achieves sample-efficient Pareto front discovery while supporting constraints and parallel evaluations. The approach is demonstrated on a synthetic ZDT4 benchmark and a sustainable bioethanol process, where MOBONS outperforms baseline MOBO methods in hypervolume progression and provides valuable local sensitivity insights around Pareto-optimal designs. Overall, MOBONS offers a practical, scalable pathway to greener profits and smarter designs by leveraging the intrinsic structure of complex engineering systems.

Abstract

Designing modern industrial systems requires balancing several competing objectives, such as profitability, resilience, and sustainability, while accounting for complex interactions between technological, economic, and environmental factors. Multi-objective optimization (MOO) methods are commonly used to navigate these tradeoffs, but selecting the appropriate algorithm to tackle these problems is often unclear, particularly when system representations vary from fully equation-based (white-box) to entirely data-driven (black-box) models. While grey-box MOO methods attempt to bridge this gap, they typically impose rigid assumptions on system structure, requiring models to conform to the underlying structural assumptions of the solver rather than the solver adapting to the natural representation of the system of interest. In this chapter, we introduce a unifying approach to grey-box MOO by leveraging network representations, which provide a general and flexible framework for modeling interconnected systems as a series of function nodes that share various inputs and outputs. Specifically, we propose MOBONS, a novel Bayesian optimization-inspired algorithm that can efficiently optimize general function networks, including those with cyclic dependencies, enabling the modeling of feedback loops, recycle streams, and multi-scale simulations - features that existing methods fail to capture. Furthermore, MOBONS incorporates constraints, supports parallel evaluations, and preserves the sample efficiency of Bayesian optimization while leveraging network structure for improved scalability. We demonstrate the effectiveness of MOBONS through two case studies, including one related to sustainable process design. By enabling efficient MOO under general graph representations, MOBONS has the potential to significantly enhance the design of more profitable, resilient, and sustainable engineering systems.

Figures (9)

• Figure 1: Illustration of a complex network system integrating process simulation DWSIM, CFD openfoam, life cycle analysis (LCA), ecological modeling iTree, and economic evaluation. The design variables (highlighted in red) impact reactor sizing, material properties, and urban planning, while the models interact dynamically to assess environmental and economic trade-offs. The figure serves as a motivating example, highlighting the challenges inherent in optimizing such systems. We aim to develop algorithms capable of tackling a broad class of these problems, which often involve high-fidelity partial differential equation solvers, embedded optimization modules, and dynamic feedback loops.
• Figure 2: Illustrative examples of acyclic (left) and cyclic (right) function networks. The acyclic graph must be capable of being topologically ordered so that future nodes only depend on previous nodes, which simplifies the evaluation. The cyclic graph, on the other hand, may have dependencies that come in the form of equality constraints, which significantly complicates determination of statistical properties of the network when the node functions are modeled as Gaussian processes.
• Figure 3: Comparison of multi-objective optimization algorithms on the ZDT4 problem, showing hypervolume progression over iterations. The dashed black line indicates the estimated maximum hypervolume from the reference point. MOBONS reaches the maximum hypervolume faster than all baseline methods, demonstrating its effectiveness in leveraging function network structure when selecting candidates.
• Figure 4: Comparison of function evaluations for each algorithm on the ZDT4 problem, showing the distribution of evaluated points over the two objectives. Each subplot presents the median performing replicate across 30 independent runs. MOBONS successfully pushes samples toward the true Pareto front, demonstrating superior efficiency in balancing exploration and exploitation. qEHVI shows limited exploration, leading to a clustered sampling pattern, while qPOTS and Random fail to fully capture the Pareto-optimal region.
• Figure 5: Process diagram for the ethanol fermentation system, consisting of a fermentation reactor followed by two distillation columns. The optimization goal is to balance economic and environmental performance by adjusting process design variables.

Theorems & Definitions (1)

• Remark 1