New Bayesian optimisation framework for robust multi–objective design: decoupling performance and uncertainty

Robust optimisation is increasingly critical in engineering design due to the growing emphasis on robustness under uncertainty. However, existing methods often neglect explicit control of performance stability, particularly regarding the variance of objectives. To address this limitation, this work introduces a Variance–constrained Robust Bayesian Optimisation framework to efficiently perform multi–objective optimisation considering input uncertainty. A robust Gaussian process is employed to quantify input uncertainty by providing both the expected objective value and its associated uncertainty. These two aspects are then integrated using a variance–penalised scalarisation method, which leverages user–defined parameters to flexibly balance mean performance and variability, thereby accommodating different levels of stability tolerance. During optimisation, this scalarisation steers the acquisition function toward regions consistent with the desired stability preference; at reporting time, surrogate estimates are used to discard any designs that violate the uncertainty threshold, and the Pareto set is computed from the remaining stability–feasible points. The novelty lies in enabling a flexible, context–dependent design selection, while preserving the data–efficiency advantages of Bayesian Optimisation. The framework has been thoroughly tested on a series of synthetic benchmark problems, including a higher–dimensional (6D) case, demonstrating its effectiveness in handling varying uncertainty tolerances and substantiating its strong potential for real–world applications.