Categorical Optimization with Bayesian Anchored Latent Trust Regions for Structural Design under High-Dimensional Uncertainty
Authors: Zhangyong Liang, Jie Hou, Huanhuan Gao, Manyu Xiao
Organizations: National Center for Applied Mathematics, Tianjin University, Weijin Road 92, 300072, Tianjin, China · School of Mechanical Engineering, Northwestern Polytechnical University, Youyixi Road 127, 710072, Xi’an, China · School of Mechanical and Aerospace Engineering, Jilin University, Renmin Street 5988, 130025, Changchun, China · Xi’an Key Laboratory of Scientific Computation and Applied Statistics, School of Mathematics and Statistics, Northwestern Polytechnical University, Youyixi Road 127, 710072, Xi’an, China
Abstract
Categorical structural optimization under aleatoric uncertainty is challenging because each design variable must be selected from a finite catalog of admissible instances, while each candidate design may require expensive stochastic finite-element evaluations. Existing latent-space optimization strategies can reduce the dimensionality of catalog attributes, but they often treat the reduced space as a continuous search domain. The resulting continuous optimum must then be rounded off to a nearby catalog instance, which may alter the objective value, constraint status, or physical interpretation of the design. To address this issue, this paper proposes the \textbf{C}ategorical \textbf{O}ptimization with \textbf{B}ayesian \textbf{A}nchored \textbf{L}atent \textbf{T}rust Regions (\textbf{COBALT}) framework for high-dimensional categorical Optimization Under Uncertainty. COBALT first embeds the physical catalog into a low-dimensional latent representation and locks the mapped instances as a discrete anchored graph. A data-independent random tree decomposition is then used to provide bounded-complexity additive modeling over high-dimensional categorical variables. On this anchored domain, an additive SAAS-GP surrogate is fitted to heteroscedastic MC-FEA observations, and a trust-region discrete graph acquisition search selects the next admissible catalog configuration without continuous relaxation or rounding-off. The proposed strategy is applied to robust design optimization of complex bar structures, considering structural weight, strain energy, and local buckling performance. By evaluating only valid catalog designs through the MC-FEA oracle, COBALT preserves physical admissibility throughout the active learning loop and improves the efficiency of robust categorical structural optimization.
Expensive constrained optimization problems in real-world industry design often involve constraint thresholds that are difficult to determine in advance. Engineers may need to adjust constraint thresholds to explore different feasibility-performance trade-offs, requiring solutions under a wide range of threshold settings. However, existing constrained Bayesian optimization methods treat each threshold configuration independently, leading to repeated optimization and failing to exploit the shared relationship among continuously varying thresholds. To address this challenge, we propose constraint-bound agnostic Bayesian optimization (CBA-BO), a learning-based framework that learns a parametric constraint model mapping thresholds to optimal solutions. Once learned, CBA-BO directly predicts solutions for arbitrary unseen threshold configurations without additional optimization, with a one-step Bayesian optimization refinement further improving solution quality. Experiments on benchmark and engineering problems demonstrate that CBA-BO learns a transferable threshold-solution mapping, enabling efficient prediction and optimization for arbitrary threshold queries. An intent-guided constraint-bound recommendation mechanism is further developed to improve objective performance while satisfying user-specified constraint preferences.
Mixed categorical-continuous optimization arises in many practical domains, yet remains challenging. In the black-box setting, evolution strategy-based approaches have shown promise in extending the efficiency and robustness of the CMA-ES to mixed-variable spaces. However, these methods exhibit worsened performance when strong categorical-continuous interactions are present, as their underlying search distributions assume independence between categorical and continuous variables. To address this limitation, we propose a bilevel optimization framework that explicitly captures such interactions by optimizing over categorical variables in an outer loop, and over continuous variables conditioned on each categorical configuration in an inner loop. We formulate each level of the bilevel problem as a stochastic relaxation under information-geometric optimization. To mitigate the high computational cost inherent to bilevel optimization, we introduce a warm-starting strategy that accelerates the lower-level search by selecting the best among multiple cached configurations and updating the cache after each iteration. Experimental results on binary-continuous domain demonstrate that the proposed method outperforms existing state-of-the-art approaches in interaction-handling capability while also being more computationally efficient across benchmarks encompassing both previously reported and newly proposed types of interaction.
Many design tasks can be cast as black-box function optimization, enabling use of Bayesian optimization to find an ideal design with minimal number of trials. However, often we do not actually need the optimum but instead a sufficiently good solution is enough, for instance a material that is durable enough for its intended use. In most cases there are multiple satisfactory solutions, forming a superlevel set of the function, raising a key question of which one to prefer. We answer this by explaining why robustness to input perturbations that may occur when the solution is deployed is a good criterion and by introduce a Bayesian optimization method that efficiently finds satisficing solutions that are robust to maximally large perturbations. In contrast to previous works, we assume the inputs can be accurately controlled during optimization, but will be perturbed after the deployment.
Samuli Kinnunen, Petrus Mikkola, Antti Niskanen +1