Swine Diet Design using Multi-objective Regionalized Bayesian Optimization

TL;DR

The paper tackles swine diet design as a multi-objective optimization problem involving $f_c(x)=\mathbf{c}^T\mathbf{x}$ (cost), $f_l(x)=\mathbf{l}^T\mathbf{x}$ (lysine), and $f_e(x)=\mathbf{e}^T\mathbf{x}$ (energy) under material and nutritional constraints. It advances Multi-objective Regionalized Bayesian Optimization (MORBO), which partitions the high-dimensional search space into multiple trust regions with local GP surrogates and coordinated exploration, improving diversity of the Pareto front over standard MOBO. Empirical results show MORBO achieves more diverse non-dominated solutions and better early domination of the reference MFP method, while MOBO can reach higher hypervolume but with less front diversity; batching MORBO further accelerates convergence. The work demonstrates MORBO’s practical potential to accelerate swine diet formulation under complex, multi-source data and supports more informed decision-making for sustainable, cost-effective feeding strategies, with future work focusing on kernel choice and incorporating contextual, farm-level variability.

Abstract

The design of food diets in the context of animal nutrition is a complex problem that aims to develop cost-effective formulations while balancing minimum nutritional content. Traditional approaches based on theoretical models of metabolic responses and concentrations of digestible energy in raw materials face limitations in incorporating zootechnical or environmental variables affecting the performance of animals and including multiple objectives aligned with sustainable development policies. Recently, multi-objective Bayesian optimization has been proposed as a promising heuristic alternative able to deal with the combination of multiple sources of information, multiple and diverse objectives, and with an intrinsic capacity to deal with uncertainty in the measurements that could be related to variability in the nutritional content of raw materials. However, Bayesian optimization encounters difficulties in high-dimensional search spaces, leading to exploration predominantly at the boundaries. This work analyses a strategy to split the search space into regions that provide local candidates termed multi-objective regionalized Bayesian optimization as an alternative to improve the quality of the Pareto set and Pareto front approximation provided by BO in the context of swine diet design. Results indicate that this regionalized approach produces more diverse non-dominated solutions compared to the standard multi-objective Bayesian optimization. Besides, the regionalized strategy was four times more effective in finding solutions that outperform those identified by a stochastic programming approach referenced in the literature. Experiments using batches of query candidate solutions per iteration show that the optimization process can also be accelerated without compromising the quality of the Pareto set approximation during the initial, most critical phase of optimization.

Figures (7)

• Figure 1: General scheme of the proposed methodology. Image adapted from guribe2024.
• Figure 2: Convergence of MORBO process for HV values across iterations and for different variations of $n_{TS}$. Solid lines represent the mean value for each $n_{TS}$, and shaded areas represent the mean $\pm$ the standard deviation per $n_{TS}$.
• Figure 3: Boxplots of HV obtained by MORBO for different hyperparameter configurations during 30 executions. (A) shows boxplots for $n_{TR} = 3$; (B) shows boxplots for $n_{TR} = 5$; and (C) shows boxplots for $n_{TR} = 8$.
• Figure 4: Figure (A) illustrates the average distances between consecutive solutions in the ingredient vector space. Figure (B) presents the average distances between consecutive solutions estimated in the nutrient vector space, while Figure (C) displays the average distances between consecutive solutions in the objective space. The distances are computed over 50 iterations for all figures and for both the MOBO and MORBO methods.
• Figure 5: UMAP projections of Pareto sets and Pareto fronts obtained by MOBO and MORBO. The solution provided by pena2009multiobjective is also included for the sake of comparison. The color bar represents the cumulative variance for the three objectives obtained from their corresponding GP surrogate models at the time when each of the solutions was suggested as the new query point during the BO process. Each marker corresponds to solutions that outperform the one obtained in pena2009multiobjective for each objective: cost (C), Lysine (L), energy (E), pairs of objectives: Cost-Lysine (CL), Cost-Energy (CE), Lysine-Energy (LE), and all three objectives combined, Cost-Lysine-Energy (CLE).