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

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.

💡 Research Summary

The paper addresses the pressing need to simultaneously optimize economic performance, resilience, and environmental sustainability in modern industrial system design. Traditional multi‑objective optimization (MOO) approaches fall into two camps: white‑box methods that require explicit equations and become intractable for large, multi‑physics models, and black‑box methods that treat the entire process as an opaque function, demanding many expensive evaluations. Grey‑box methods have attempted to bridge the gap, but they usually assume a sequential or hierarchical mapping of sub‑models, which does not reflect the bidirectional, cyclic interactions common in real process networks.

To overcome these limitations, the authors propose a unifying grey‑box framework based on a graph (network) representation. The system is decomposed into K function nodes f₁…f_K, each receiving a subset of the design variables x and the outputs of other nodes as inputs. Formally, y_k = f_k(x_{I(k)}, {y_j}_{j∈J(k)}). The directed edges define dependencies, and cycles are explicitly allowed, enabling the modeling of recycle streams, feedback loops, and multi‑scale coupling (e.g., a CFD heat exchanger feeding back to a flow‑sheet model). The overall objective vector G(x) is expressed as a linear projection of the node outputs, G(x)=C Y* (x), where Y* (x) is a solution of the implicit fixed‑point system Y = F(x, Y).

Instead of placing a Gaussian Process (GP) prior directly on the objective functions, MOBONS (Multi‑Objective Bayesian Optimization for Networked Systems) places independent GP priors on each node function f_k. This structural surrogate exploits the known network topology: intermediate node outputs observed during simulation are incorporated into the GP posteriors, dramatically increasing the information gained per expensive evaluation. The algorithm proceeds as follows:

1. Posterior Update – Using all previously observed (input, output) pairs for each node, compute the GP posterior distributions.
2. Network Evaluation – For a candidate design x, solve the implicit network equations (via fixed‑point iteration or variational inference) to obtain a sample of Y* (x).
3. Acquisition Generation – Employ Thompson sampling to draw a realization of each node GP, propagate it through the network, and evaluate a multi‑objective acquisition function. The acquisition problem is solved with a multi‑objective evolutionary algorithm (a NSGA‑II variant), allowing simultaneous proposal of a batch of candidate designs.
4. Parallel Evaluation – The batch is evaluated in parallel on high‑performance computing resources; observed intermediate and final outputs are added to the data set.
5. Constraint Handling – Inequality constraints are incorporated either as additional GP‑modeled functions with penalty terms in the acquisition or via an ε‑constraint formulation.

Key innovations of MOBONS include:

• Cyclic Graph Support – Unlike prior graph‑based BO methods that restrict to directed acyclic graphs, MOBONS can handle cycles, making it suitable for realistic process networks.
• Node‑Level Surrogates – By modeling each sub‑function separately, the method leverages partial observations, reducing the number of full‑system evaluations needed.
• Thompson Sampling + Evolutionary Search – This combination balances exploration and exploitation efficiently in high‑dimensional design spaces and naturally yields a diverse set of Pareto‑optimal candidates.
• Batch Parallelism – The algorithm is designed for concurrent evaluations, cutting wall‑clock time in practice.

The authors validate MOBONS on two case studies. The first is a conventional chemical process design problem where MOBONS achieves comparable Pareto coverage with roughly 40 % fewer expensive function calls than standard multi‑objective BO (e.g., TSEMO) and classic NSGA‑II. The second case focuses on a sustainable process design involving energy consumption and carbon emissions. The underlying network includes a recycle loop and multi‑scale coupling between a CFD heat exchanger model and a process flow‑sheet. Here, MOBONS outperforms BOFN (a recent graph‑based BO that assumes acyclicity) by producing a Pareto front that is more than twice as extensive, clearly revealing trade‑offs between profit and environmental impact.

In conclusion, the paper demonstrates that embedding explicit network structure into a grey‑box Bayesian optimization framework yields substantial gains in sample efficiency, scalability, and solution quality for expensive, interconnected industrial simulations. The approach opens avenues for further research, such as robust optimization under adversarial uncertainty, dynamic adaptation to changing operating conditions, and integration with advanced constraint‑handling techniques.