Highlights Categorical Optimization with Bayesian Anchored Latent Trust Regions for Structural Design under HighDimensional Uncertainty Zhangyong Liang, Huanhuan Gao • COBALT tackles high-dimensional categorical OUU with costly MC-FEA. • COBALT locks latent catalog instances as discrete physical anchors.
arXiv:2604.25241v1 [cs.LG] 28 Apr 2026
• Additive SAAS-GP infers sparse effects under heteroscedastic noise. • Trust-region graph acquisition returns valid designs without rounding. • Random tree decomposition enables efficient high-dimensional OUU.
Categorical Optimization with Bayesian Anchored Latent Trust Regions for Structural Design under High-Dimensional Uncertainty Zhangyong Lianga , Huanhuan Gaob,c, a National Center for Applied Mathematics Tianjin University Tianjin 300072 China b Key Laboratory of CNC Equipment Reliability, Ministry of Education\National Key Laboratory of Automotive Chassis Integration and Bionics,
School of Mechanical and Aerospace Engineering, Jilin University, Renmin Street 5988, 130025, Changchun, China c School of Mechanical and Aerospace Engineering, Jilin University, Renmin Street 5988, 130025, Changchun, 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 Categorical Optimization with Bayesian Anchored Latent Trust Regions (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 roundingoff. 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. Keywords: Categorical optimization, Optimization under uncertainty, Manifold learning, Bayesian optimization, Gaussian process.
1. Introduction In various optimization problems, design variables are generally classified as continuous, discrete (including integer), and categorical ones [1–4]. A categorical variable is distinguished by the fact that its admissible values are finite instances, or levels, selected from a predefined category rather than arbitrary real numbers. A typical structural example is the selection of a bar cross-section from a catalog of standard profiles. These catalog sections may differ simultaneously in shape, area, moments of inertia, and section modulus, so their physical meaning cannot be faithfully represented by a single scalar index. Classical methods [5–10] for optimization problems with categorical variables often transform the available instances into artificial real-valued codes or binary strings and then employ evolutionary optimizers such as genetic algorithms. Such encodings are convenient for general-purpose search and do not require gradient information. However, the numerical distance between two encoded values usually has no direct relation to the physical difference between the corresponding catalog instances. This loss of physical meaning may reduce the search efficiency and obscure the interpretation of the optimized structural design. Email address: [email protected] (Huanhuan Gao)
Other works have been proposed to treat categorical variables in a more specific manner [11–13]. For example, when dealing with unordered categorical variables, simplex coding is broadly applied, which suits evolutionary methods well [1, 14, 15]. In this coding, the distances between any two design instances are kept equal, thus forming a regular simplex in the space, whose dimensionality is always one less than the number of instances. As a result, if the number of instances is large, the dimensionality of the related space becomes computationally prohibitive. To avoid assigning arbitrary labels to physically different sections, a more informative strategy is to represent each categorical instance by a multi-dimensional discrete vector of physical attributes. Based on this multi-dimensional representation, manifold learning techniques can be utilized to discover the lowerdimensional intrinsic manifold embedded in the higher-dimensional attribute space of the catalog. The main linear manifold learning techniques are Principal Component Analysis (PCA) [16] and Multi-Dimensional Scaling (MDS) [17]. By applying eigenvalue decomposition, PCA tries to preserve the most covariance information in the reducedorder space, while MDS focuses on the preservation of Euclidean distances between sample points. However, they lack the ability to map a non-linear manifold (e.g., the “Swiss Roll”) to a reduced-order space [18]. Typical non-linear manifold learning techniques include Locally Linear Embedding (LLE) [19, 20], Kernel PCA (KPCA) [21, 22], and Isomap [18, 23]. Particularly, Isomap is a variant of MDS that uses geodesic distances calculated by the Dijkstra algorithm [24] instead of Euclidean distances, thereby preserving the intrinsic non-linear topology of the instance set. These manifold learning techniques have been widely applied in identification problems [25], structural optimization [26], and visualization [27]. In previous deterministic optimization frameworks, the reduced-order representation was further approximated by continuous polynomial interpolations so that gradient-based methods, such as the Method of Moving Asymptotes (MMA) [28, 29], could be applied on a continuous manifold. Since the original categorical problem admits only finite catalog instances, the continuous optimum must then be rounded off to a nearby admissible instance through a nearest-neighbor search [30]. This continuous relaxation and rounding-off strategy is efficient in the continuous search stage, but the rounded design may differ from the continuous solution in objective value, constraint status, and physical interpretation. We refer to this mismatch as Decoding Failure (or Rounding-off Error), which becomes more pronounced when the catalog manifold is highly non-linear or sparsely sampled. The above difficulty is further amplified when structural design is performed under aleatoric uncertainty, such as manufacturing tolerances, material property fluctuations, and environmental load variations. Transitioning from deterministic categorical optimization to Optimization Under Uncertainty (OUU) requires evaluating robust performance metrics through stochastic simulations, for example Monte Carlo-based Finite Element Analysis (MC-FEA). Each admissible catalog combination is then associated with expensive statistical estimates rather than a single deterministic response, and the resulting observations are generally noisy and heteroscedastic. This makes direct enumeration, deterministic continuous optimizers, and traditional heuristic global searches [31, 32] inefficient for high-dimensional categorical OUU. Bayesian Optimization (BO) [33, 34] provides a principled surrogate-assisted framework for expensive and noisy evaluations. By maintaining a probabilistic surrogate model, typically a Gaussian Process (GP) [35, 36], BO balances exploration and exploitation while propagating predictive uncertainty throughout the optimization loop [37]. Nevertheless, standard BO scales poorly with high-dimensional combinatorial spaces. Although dimensionality-reductionenhanced BO methods (i.e., Latent Space BO) have been proposed [38, 39] to embed the high-dimensional search space into a lower-dimensional latent space, existing frameworks often optimize a continuous acquisition function in the latent space. Consequently, when the feasible designs must remain strict catalog instances, these methods still require a decoding or rounding-off step and remain vulnerable to the mismatch described above. To address the decoding failure and dimensionality challenges in high-dimensional categorical OUU, this paper proposes the Categorical Optimization with Bayesian Anchored Latent Trust Regions (COBALT) framework, which integrates discrete manifold anchoring with Bayesian optimization under uncertainty. The main contributions of this work are summarized as follows: • From Continuous Relaxation to Absolute Discrete Anchoring. COBALT does not treat the dimensionalityreduced latent space as a searchable continuum. Instead, it locks the mapped catalog instances as a discrete network of physically admissible anchor points. The optimizer is therefore restricted to valid categorical designs throughout the search. • Data-Independent Random Tree Decomposition. COBALT avoids learning a fragile interaction structure 2
from sparse and noisy observations. It samples random tree decompositions over the categorical variables. This provides a bounded-complexity view of high-dimensional catalog combinations and repeatedly varies the pairwise interactions modeled by the surrogate. • Uncertainty-Aware Sparse Inference. Multi-component selections, such as cross-section assignments for different bar groups, lead to rapid combinatorial growth. MC-FEA further introduces heteroscedastic observation noise. To address these difficulties, we integrate Sparse Axis-Aligned Subspace (SAAS) priors into the Gaussian Process. Under fully Bayesian inference, the surrogate identifies influential latent features and variable interactions while remaining robust to noisy observations. • Discrete Graph Acquisition Maximization. COBALT does not optimize a continuous acquisition function followed by rounding-off to the catalog. Instead, graph-based evolutionary operators based on Dijkstra’s shortest paths are employed within the Bayesian active learning loop. The search traverses the discrete anchored graph inside dynamically scaled latent trust regions. Thus, the recommended design remains a physically admissible categorical configuration before the MC-FEA oracle is called. • Robust MC-FEA Evaluation of Valid Catalog Designs. COBALT separates deterministic catalog geometry from aleatoric structural uncertainty. The catalog geometry is fixed by the anchored latent graph. The aleatoric uncertainty is evaluated only after an admissible categorical configuration has been selected. This allows the MC-FEA oracle to estimate robust objectives and constraints without introducing additional decoding ambiguity. We organize the contents of this paper as follows. Section 1 gives the research backgrounds and motivation. In Section 2, we formulate the robust categorical structural optimization problem under uncertainty. Section 3 details the proposed COBALT framework, including the absolute anchoring of discrete latent manifolds, data-independent random decompositions, SAAS-GP surrogate modeling, and trust-region discrete graph acquisition. Section 4 presents numerical tests involving complex structures under uncertainty to demonstrate the superiority of the proposed framework. Finally, conclusions and prospects are drawn in Section 5. 2. Problem formulation 2.1. Categorical structural optimization under uncertainty The values which a categorical variable can take are called instances, or levels [40]. In categorical structural optimization, these instances usually correspond to available engineering choices such as standard cross-section types selected from a design catalog. Each instance is described by a multi-dimensional discrete vector, and the components of this vector are the physical attributes of the corresponding design choice. Therefore, an available catalog can be represented as a finite set of points in a multi-dimensional attribute space. In the deterministic categorical formulation, the objective and constraints are evaluated once for every selected combination of instances. However, real-world structural systems are subjected to aleatoric uncertainties, such as manufacturing tolerances, material property fluctuations, and unpredictable environmental loads. Thus, the deterministic categorical optimization problem must be extended to Optimization Under Uncertainty (OUU), where the same discrete instance combination is assessed by its statistical structural performance. As illustrated in Fig. 1, even for a fixed I-beam instance selected from the catalog, its geometric attributes such as flange width, flange thickness, and section height may fluctuate around their nominal values due to manufacturing uncertainty. Let ξ ∼ P(ξ) denote the random parameter vector characterizing these inherent uncertainties. For the sake of generality, we present the robust constrained categorical optimization problem with e categorical variables as follows: q min.: Jrobust (x1 , x2 , · · · , xe ) = Eξ [J(x1 , x2 , · · · , xe , ξ)] + γ Vξ [J(x1 , x2 , · · · , xe , ξ)]; q s. t.: Grobust (x1 , x2 , · · · , xe ) = Eξ [g(x1 , x2 , · · · , xe , ξ)] + β ◦ Vξ [g(x1 , x2 , · · · , xe , ξ)] ≤ 0; (1) xi ∈ Xi = {x1i , x2i , · · · , xni }, i = 1, 2, · · · , e; xji = (1 aji , 2 aji , · · · , M aji )T , j = 1, 2, · · · , n. 3
Figure 1: The instance geometric parameters subordinate to the Gaussian distribution.
In Eq. (1), Jrobust is the robust objective function obtained from the poriginal structural performance J. It is written as a weighted sum of the expectation Eξ [·] and the standard deviation Vξ [·], where γ ≥ 0 controls the robustness penalty associated with performance fluctuation. Similarly, Grobust denotes the robust constraint functions, and the reliability index vector β controls the strictness of the constraint margin (◦ denotes the Hadamard product). xi denotes the i-th categorical variable, and its value must be selected from the finite instance set Xi . xji is the j-th available instance of the i-th variable, represented by an M-dimensional attribute vector. l aji is the l-th physical attribute of this instance, such as area, moment of inertia, section modulus, or other catalog descriptors. For any fixed categorical combination (x1 , x2 , · · · , xe ), the statistical moments in Eq. (1) are estimated by repeatedly evaluating the assembled structural model under sampled uncertainty. This evaluation typically relies on Monte Carlobased Finite Element Analysis (MC-FEA). Because only a finite number of samples can be used, the exact robust objective Jrobust is not observed directly. Instead, the optimizer receives an expensive noisy observation yobs : yobs (x1 , x2 , · · · , xe ) = Jrobust (x1 , x2 , · · · , xe ) + ϵ,
ϵ ∼ N(0, σ2noise )
(2)
where ϵ represents the heteroscedastic noise induced by stochastic simulation and finite Monte Carlo sampling. The search domain is therefore a finite but massive categorical set containing up to ne possible combinations. This discreteness is fundamentally different from a continuous structural design space: a candidate is admissible only when every variable is assigned one of its available instances. Combined with the expensive noisy evaluation in Eq. (2), this combinatorial structure makes direct enumeration, deterministic continuous descent, and purely heuristic global search inefficient. These bottlenecks motivate a sample-efficient Bayesian Optimization framework that explicitly respects the categorical nature of the design variables. 2.2. Bayesian Optimization In this paper, Bayesian Optimization (BO) is used to guide the search for the best robust categorical design under a limited evaluation budget. As will be formally defined in Section 3, each categorical combination is represented by latent coordinates Z ∈ RD obtained from the selected catalog instances. BO sequentially minimizes the expensive black-box function f (Z) = Jrobust (Z) by iterating two key steps: fitting a surrogate model from evaluated categorical designs and selecting the next admissible design through an acquisition function. Gaussian Process Surrogate: To approximate the robust response over the catalog combinations with few MCFEA evaluations, we choose the Gaussian Process (GP). Following [41], a GP can be interpreted from a weightspace view. Given a set of t observations over the latent representation of evaluated categorical designs, denoted by Dt = {(Z(i) , y(i) obs ) | i = 1, ..., t}, we build the surrogate as follows. Denoting all inputs as Z1:t and the output vector as yt , we first consider a linear surrogate: f (Z) = ϕ(Z)T w,
yobs = f (Z) + ϵ,
ϵ ∼ N(0, σ2noise )
(3)
where w is the model parameter vector, ϕ(Z) is the feature mapping, and ϵ is the observation noise. By taking a Bayesian treatment and placing a zero-mean Gaussian prior on the weight vector w ∼ N(0, Σ p ), the predictive distribution can be analytically derived. Since the feature mappings always appear in the form of an inner product with 4
respect to Σ p , this implies we are comparing inputs in a feature space and enables us to apply the kernel trick. Therefore, instead of explicitly defining a feature mapping ϕ(·), we define a covariance kernel k(Z, Z′ ) = ϕ(Z)T Σ p ϕ(Z′ ). Substituting it into the predictive distribution gives the vanilla GP posterior for an unseen input Z∗ : f∗ | Z1:t , yt , Z∗ ∼ N(µt (Z∗ ), σ2t (Z∗ )), where µt (Z∗ ) = kTt (Z∗ )[Kt + σ2noise I]−1 yt ,
(4)
σ2t (Z∗ ) = k(Z∗ , Z∗ ) − kTt (Z∗ )[Kt + σ2noise I]−1 kt (Z∗ ). where Kt is the covariance matrix evaluated on Dt , and kt (Z∗ ) contains the kernel evaluations between Z∗ and Dt . From this interpretation, the GP compares categorical designs through their latent representations in the Reproducing Kernel Hilbert Space (RKHS) defined by the kernel, while the original feasibility of the design is still determined by the underlying catalog instances. Acquisition Function Optimization: Given the GP posterior, the next step is to decide which categorical combination should be evaluated. The exploitation and exploration balance is achieved by designing an acquisition function α(Z|Dt ). Though numerous acquisition functions exist [42], we adopt the Lower Confidence Bound (LCB) acquisition for minimization tasks: αLCB (Z|Dt ) = µt (Z) − κσt (Z), (5) where κ is a hyper-parameter controlling the exploration level. 2.3. High-Dimensional BO with Decompositions BO in high-dimensional spaces is an active area of research. In categorical structural optimization, the dimensionality increases with the number of grouped members or components whose cross-sections must be selected from catalogs. Decomposition-based strategies are therefore introduced to model a large categorical combination as several lower-dimensional subproblems. Specifically, a decomposition g over e design components is a collection of sets c, called sub-components, consisting of categorical variable indices i ∈ {1, 2, . . . , e}. Decomposition-based BO assumes P that the black-box function decomposes according to g: f (Z) = c∈g fc Z[c] , where Z[c] extracts the latent attributes corresponding only to the categorical variables that appear in index set c. Additive GP Kernels: Compared to standard BO, decomposition methods employ additive kernels [43, 44] that P better suit a decomposable black-box function: kg (Z, Z′ ) = c∈g kc Z[c] , Z′[c] . If two categorical variables i and j do not appear together in any of the sets c, the kernel does not model their interaction. [45] showed that the posterior of each component subfunction fc Z[c] follows an additive distribution p fc Z[c] |Dt = N µt,c Z[c] , σ2t,c Z[c] , where: µt,c Z[c] = kTt,c Z[c] [Kt + σ2noise I]−1 yt σ2t,c Z[c] = kc Z[c] , Z[c] − kTt,c Z[c] [Kt + σ2noise I]−1 kt,c Z[c] , where kt,c (Z[c] ) is a vector of kernel evaluations between Z and all inputs in Dt while only considering those variables i ∈ c that appear in c. If the size of each set c is strictly smaller than the total number of categorical variables e, the surrogate is fitted on lower-dimensional subspaces rather than on the full catalog product at once. Furthermore, the acquisition function P natively inherits an additive structure: α(add-LCB) (Z) = c∈g µt,c (Z[c] ) − κσt,c (Z[c] ) . However, this advantage depends t on the selected decomposition g. Existing methods attempt to learn g from data using maximum likelihood, but sparse noisy observations from structural OUU may not reliably reveal the true interaction pattern among categorical variables. We expand on these challenges and detail the corresponding resolution in Section 3. 3. Methodology To fundamentally overcome the decoding failures induced by continuous relaxation and the computational intractability of Optimization Under Uncertainty (OUU) in high-dimensional categorical spaces, we propose the Categorical Optimization with Bayesian Anchored Latent Trust Regions (COBALT) framework. The method starts from the same 5
COBALT: Active Learning Loop for Categorical Structural Optimization Discrete Manifold Anchoring
Random Decomposition
Physical Cross-Section Catalog C
Trust-Region Acquisition
e Categorical Variables v3
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Random spanning tree Et ∼ Uniform(Gtree ) ∑ kGt (Z, Z′ ) = (u,v)∈Et ( ) ′ kuv yu , yv ; yu , yv′
Znext = arg 3
Info-gain bound γT = O(e(log T ) )
Anchor to discrete grid ΩD ; reject continuous relaxation
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Stochastic Structural MC-FEA Oracle Jrobust (Z) = Eξ [J(Z, ξ)] + γ yobs = Jrobust (Znext ) + ϵ,
√
Vξ [J(Z, ξ)]
ϵ ∼ N (0, σn2 )
Evaluates exact discrete structure under aleatoric uncertainty ξ to close the active learning loop.
Figure 2: The active learning loop of the proposed COBALT framework. The physical catalog is embedded and locked as the discrete latent grid ΩD . A random tree decomposition and SAAS-GP surrogate guide trust-region acquisition over admissible anchored designs, while MC-FEA evaluates the selected configuration under aleatoric uncertainty and updates the Bayesian loop.
categorical design description used in deterministic structural optimization: each design variable can only take one of the available catalog instances, and each instance is represented by a vector of physical section attributes. Different from the traditional two-stage strategy that first creates a continuous manifold and then rounds the continuous solution back to the nearest available section, COBALT treats the discrete nature of categorical optimization as a hard constraint throughout the entire process. The methodology is organized into four modules, as illustrated in Fig. 2: Specifically, COBALT first employs non-linear manifold embedding (Isomap) to arrange the available physical section instances in a low-dimensional latent space and then locks these mapped instances as a discrete tensor grid ΩD . It then introduces random decomposition to describe the interaction pattern among many categorical variables without learning a fragile structure from sparse data. Building on this anchored discrete catalog space, an additive SAAS-GP surrogate together with graph-based evolutionary operators maximizes the LCB acquisition only over admissible anchored instances within a dynamically scaled trust region T Rt , selecting the next design Znext without any continuous relaxation or rounding-off. Finally, the selected categorical configuration is assembled into the structural model and physically evaluated through MC-FEA under aleatoric uncertainty ξ to update the surrogates and close the active learning loop. 3.1. Dimensionality reduction and discrete latent manifold anchoring As defined in Section 2, the available values of any categorical design variable x form a finite catalog. Each catalog instance is described by a multi-dimensional attribute vector: X = [x1 , x2 , · · · , xn ], xj = (1 aj , 2 aj , · · · , M aj )T , j = 1, 2, · · · , n, 6
(6)
where X is an M × n matrix and xj is the j-th available instance. In structural applications, the entries of xj may be the area, moments of inertia, section modulus, or other physical descriptors of a catalog section. To obtain a compact ordering of these discrete instances, we employ non-linear manifold learning (e.g., Isomap) to find an optimal mapping Φ : RM → Rm (M > m), projecting the high-dimensional attributes to a low-dimensional representation: Φ
X− → Y = [y1 , y2 , · · · , yn ], yj = (1 bj , 2 bj , · · · , m bj )T , j = 1, 2, · · · , n.
(7)
To preserve the relative locations of the available instances in the physical catalog, Φ minimizes the residual stress between the original high-dimensional geodesic distances and the latent distances. As in deterministic categorical optimization, the attributes are first normalized column-wise within [0, 1] when their magnitudes differ substantially, so that no single physical descriptor dominates the latent layout. The Paradigm Shift (Rejecting the Continuum): In conventional deterministic frameworks, the mapped coordinates Y are subsequently fitted with continuous polynomial interpolations to create a searchable manifold, followed by gradient-based continuous optimization (e.g., MMA). The iterative designs generated in that continuous space are generally inadmissible for the original categorical problem, because the available designs are discrete in nature. A rounding-off procedure is therefore required, and this step can change both the objective value and the constraint status when the continuous solution is projected back to a nearby catalog instance. The proposed COBALT framework removes this intermediate inadmissible stage. We mathematically lock the mapped coordinates, defining them strictly as an Absolute Anchor Set in the latent space: Acat = {y1 , y2 , · · · , yn } ⊂ Rm .
(8)
Consequently, for a structural system comprising e categorical variables, a design is formed by choosing one anchored instance for each variable. The global feasible search domain is therefore strictly confined to a discrete Cartesian product tensor grid: Z = [(y1 )T , (y2 )T , · · · , (ye )T ]T ∈ ΩD ≡ (Acat )e ⊂ RD , (9) where the total latent search dimensionality is D = m × e. Any coordinate Z < ΩD is rigorously deemed physically inadmissible. Thus, every candidate considered by COBALT corresponds to a combination of available categorical instances before any structural analysis is performed. Remark (Deterministic geometry and stochastic optimization). It may appear counter-intuitive to employ a deterministic embedding inside a Bayesian uncertainty-aware framework. The apparent tension dissolves once the heterogeneous sources of uncertainty are properly separated. The catalog X = {x1 , . . . , xn } consists of standardized cross-section instances whose physical attributes (area, moments of inertia, plastic modulus, etc.) are deterministic quantities prescribed by engineering codes (e.g., GB/T 706, AISC Steel Manual); the adjacency and relative location of these instances in the design catalog are therefore intrinsically deterministic and are captured once-and-for-all by Isomap. The genuine uncertainties of the problem are shouldered by the downstream modules through their mathematically appropriate channels: (i) the aleatoric load uncertainty ξ is absorbed into the robust scalar Jrobust via MC-FEA (Module 4); (ii) the heteroscedastic observation noise ϵ is carried by the position-dependent likelihood of the Additive SAAS-GP (Section 3.3); (iii) the epistemic uncertainty about f is propagated by the fully Bayesian NUTS posterior P( f ⋆ | Z, Dt ) and expressed through σt (Z) in the LCB acquisition; (iv) the adversarial ignorance of the additive structure g is hedged by the uniform random spanning-tree scheme S r (Section 3.2, Theorem 2). Forcing a stochastic embedding (e.g., GP-LVM) at this layer would instead pollute the anchor set Acat with posterior variance, destabilize the tensor grid ΩD , the trust region T Rt , and the Dijkstra graph used by the evolutionary acquisition operators, and ultimately invalidate the regret bound in Theorem 1, which presumes ΩD is known and fixed. The deterministic Isomap layer is therefore not a loss of generality but a structural prerequisite for preserving a fixed catalog of admissible instances while routing each uncertainty source to its proper treatment module. 3.2. Data-independent random tree decompositions Evaluating the robust objective Jrobust over the immense combinatorial catalog ΩD (e.g., ne combinations) necessitates a decomposition-based surrogate model to mitigate the curse of dimensionality. 7
Misleading Decomposition Learners: Conventional decomposition methods typically attempt to learn the optimal additive kernel structure from collected data by maximizing the marginal likelihood. However, relying entirely on local MC-FEA data to infer the interaction topology among categorical variables is highly problematic. In structural OUU, changing one catalog instance may appear to affect only a local member group in a small region of the design space, while the same replacement may trigger strongly coupled stress, displacement, or buckling responses elsewhere. Consequently, a data-driven learner can exploit this local view, falsely deduce a complete separation among variables, and become trapped in suboptimal local modes. Adversarial Formulation and Data-Independent Rules: To bypass this misleading phenomenon, we formulate the decomposition from an adversarial perspective. We postulate that the true robust objective f (·) resides in a Reproducing Kernel Hilbert Space (RKHS) H g defined by some unknown decomposition g. Instead of learning this decomposition from the currently evaluated catalog combinations, we introduce a predefined, data-independent scheme S (t) : Z+ → G that selects a decomposition graph gt from a class G at active learning round t. The capability of a UCB-style active learning algorithm operating under this scheme is theoretically governed by two competing quantities: the maximum information gain γT = maxZ I(fT , yT ) (measuring the kernel’s complexity) and the function mismatch ϵt = maxZ∈ΩD | fˆt (Z) − f (Z)| (where fˆt ∈ Ht is the closest function to f spanned by the selected kernel). The high-probability cumulative regret RT after T rounds is bounded as follows: Theorem 1. Let the objective f be selected by an adversary from an RKHS H g . With probability at least 1 − δA − δB , the cumulative regret RT of a UCB-style BO algorithm utilizing the data-independent scheme S (t) over the discrete domain ΩD incurs the following bound: hP i r p ES Tt=1 ϵt 1 , + γT + (10) RT = O T γT B + ln δA δB where B = maxt≤T ∥ fˆt ∥Ht . Theorem 1 dictates that an optimal scheme must simultaneously restrict the information gain γT and minimize the P expected mismatch ES [ Tt=1 ϵt ]. Analysing Decomposition Rules: To strictly bound γT , we restrict the decomposition class G to graphs with exclusively pair-wise components. Proposition 1. If the decomposition class G is restricted to tree-based decompositions (i.e., undirected acyclic graphs) spanning the e categorical variables, the maximum information gain is strictly bounded by γT ≤ O(e(log T )3 ), which is unconditionally smaller than incorporating fully-connected pairwise interactions. P While tree-based topologies bound the kernel complexity, minimizing the expected mismatch ES [ Tt=1 ϵt ] against an adversarial structural black-box is mathematically challenging. If S (t) constantly suggests the same fixed tree decomposition, the highly coupled structural response may repeatedly appear through omitted interactions, resulting in a constant high mismatch. Conversely, adaptively increasing the interaction complexity destroys the tree property and exponentially inflates γT . To balance these two effects, COBALT leverages the theoretical optimality of randomized sampling: Theorem 2. Let G be the class of tree-based decompositions defined over e categorical variables. For any adversarial structural function f ∈ Hg , the expected sum of mismatches across all data-independent schemes S is mathematically minimized by the scheme S r that selects a tree decomposition from G uniformly at random, i.e.: ∀S :Z+ →G
T T X X ES ϵt ≥ ES r ϵt . t=1
(11)
t=1
Motivated by Theorem 2, COBALT abandons computationally intractable and easily misled data-driven correlation learning. By uniformly sampling a random tree at each iteration, COBALT repeatedly changes the pairwise view through which catalog combinations are modeled, while keeping the complexity bounded in high-dimensional discrete spaces. 8
3.3. Uncertainty-aware additive sparse surrogate modeling (Additive-SAAS-GP) At optimization iteration t, the evaluated data consist of admissible categorical configurations and their MC-FEA t responses, denoted by Dt = {(Z(i) , y(i) obs )}i=1 . Due to the stochastic nature of uncertainty propagation, the observations yield heteroscedastic noise governed by the Monte Carlo sampling variance: (i) (i) y(i) obs = f (Z ) + ϵ ,
ϵ (i) ∼ N(0, σ2noise (Z(i) )),
(12)
where f (Z) represents the underlying true robust objective. Building upon the optimal data-independent scheme S r , we construct a fully Bayesian additive surrogate f (Z) ∼ GP(µ0 , kGt (Z, Z′ )). Based on the uniformly sampled random tree graph Gt = (V, Et ) spanning the e categorical variables (with |V| = e and |Et | = e − 1), the similarity between two categorical designs is measured by comparing the anchored latent coordinates of their selected instances. This leads to an additive kernel decomposed strictly over the tree edges: X kGt (Z, Z′ ) = (13) kuv [(yu )T , (yv )T ]T , [(y′u )T , (y′v )T ]T , (u,v)∈Et
where yu , yu ∈ Acat , and each base kernel kuv is an Automatic Relevance Determination (ARD) kernel operating exclusively on the localized 2m-dimensional latent subspace of interconnected categorical variables u and v. To further combat the combinatorial explosion and induce feature sparsity within the active pairwise subspaces, a heavy-tailed hierarchical Horseshoe prior (SAAS) is imposed on the inverse lengthscales (i.e., feature weights θd = ρ−1 d ) of the base kernels: θd ∼ HC(0, τ), τ ∼ HC(0, τ0 ). (14) ′
Under fully Bayesian inference (e.g., via the No-U-Turn Sampler, NUTS), the hyperparameter posteriors Θ = {σ2f , θ1:D , σ2noise , τ} are marginalized rather than point-estimated. The predictive posterior distribution for any unseen but admissible categorical configuration Z is rigorously integrated over the MCMC hyperparameter samples: Z ∗ P( f | Z, Dt ) = P( f ∗ | Z, Dt , Θ)P(Θ | Dt )dΘ. (15) This additive hierarchical structure forces the model to aggressively shrink the weights of irrelevant latent features while analytically decoupling the predictive mean µt (Z) and epistemic variance σ2t (Z) into pairwise summations. Consequently, the surrogate focuses on influential catalog attributes and variable interactions while remaining stable under severe heteroscedastic noise. 3.4. Trust-region discrete graph acquisition maximization Based on the decomposed predictive moments µt and σt extracted from Eq. (15), we construct the Lower Confidence Bound (LCB) acquisition function to decide which admissible catalog combination should be evaluated next. Natively inheriting the pairwise factorization over the sampled random tree graph Et , the global unconstrained acquisition is given by: X αLCB (Z) = µuv (yu , yv ) − κ · σuv (yu , yv ) , (16) (u,v)∈Et
where κ regulates the exploration trade-off. To avoid evaluating too many remote combinations in the vast ne discrete catalog space, COBALT incorporates a dynamically scaled hyper-cubic Trust Region (T Rt ) centered at the incumbent robust best design Z∗t . The trust region limits the admissible neighboring catalog combinations considered around the current best design and is mathematically defined via the Chebyshev distance (L∞ -norm): Lt , (17) T Rt = Z ∈ RD ∥Z − Z∗t ∥∞ ≤ 2 where the trust region length Lt expands after a sequence of successful improvements and contracts following consecutive evaluation failures. 9
In contrast to traditional methods utilizing continuous optimizers followed by nearest-neighbor rounding-off, identifying the optimal configuration Znext for the next expensive MC-FEA evaluation is strictly formulated as a purely discrete combinatorial optimization problem over the valid intersection set Ω(t) T R: Znext = arg min αLCB (Z), Z∈Ω(t) TR
where Ω(t) T R = Ω D ∩ T Rt .
(18)
To efficiently solve Eq. (18) over the discrete combinatorial intersection Ω(t) T R without resorting to any continuous relaxation, we repurpose graph-based evolutionary operators on the anchored catalog network as the internal acquisition maximizer. Evaluating the additively decomposed algebraic formulation αLCB (Z) incurs negligible computational cost (< 10−4 seconds) compared to an expensive MC-FEA. Therefore, the evolutionary operators can explore many alternative combinations of neighboring catalog instances as an internal surrogate search. By constraining crossover and mutation trajectories exclusively along the geodesic pathways of the discrete anchored network Acat and within the dynamic boundary T Rt , the internal engine guarantees that the identified minimizer Znext is still composed of physically feasible categorical instances. This strictly valid configuration Znext is then physically evaluated by MC-FEA without any artificial rounding-off, closing the robust Bayesian active learning loop while avoiding the inadmissible intermediate designs that appear in continuous-relaxation strategies. 3.5. Categorical Optimization Processes The preceding modules are assembled into a sequential categorical optimization process in Algorithm 1. The process contains one deterministic catalog-preparation stage and one uncertainty-aware Bayesian search stage. In the preparation stage, the physical catalog is normalized, embedded, and locked as the discrete anchor set Acat , which defines the immutable feasible tensor grid ΩD . This stage is executed only once and establishes the central invariant of COBALT: every subsequently proposed design must be an admissible combination of catalog instances rather than a relaxed point in the surrounding continuum. At each active learning iteration, COBALT samples a data-independent random tree to determine the current additive kernel structure, fits the Additive SAAS-GP surrogate to the accumulated MC-FEA observations, and constructs the LCB acquisition over the sampled tree. The trust region then restricts the candidate set to Ω(t) T R = Ω D ∩ T Rt , so acquisition maximization is performed only over physically valid anchored configurations. The selected catalog combination is evaluated by the stochastic MC-FEA oracle, appended to the dataset, and used to update both the incumbent robust optimum and the trust-region length. Thus, the algorithm separates deterministic catalog geometry, discrete interaction decomposition, Bayesian uncertainty quantification, and aleatoric physical evaluation into a closed-loop categorical optimization procedure. Algorithm 1 highlights three operational safeguards that distinguish COBALT from relaxation-based categorical optimization. First, the anchor set Acat and tensor grid ΩD remain fixed after the initial embedding, so the available instances and their adjacency relations are never changed during optimization. Second, the decomposition graph Gt is resampled independently of the observed objective values, avoiding the instability of likelihood-driven structure learners under sparse and noisy MC-FEA data. Third, the acquisition maximizer is constrained to traverse anchored graph paths inside Ω(t) T R , so the recommended point Znext is already a physically decodable design before the expensive oracle is called. The computational burden of the loop is therefore concentrated in the MC-FEA oracle rather than in acquisition maximization. Once the Additive SAAS-GP posterior is fitted, evaluating αLCB is algebraic and inexpensive, allowing the graph-based evolutionary search to explore many neighboring catalog combinations within each trust region. This design ensures that every evaluation budget is spent on structurally valid configurations and that the active learning loop improves the robust optimum without introducing rounding-off ambiguity. 4. Numerical examples To validate the effectiveness and superiority of the proposed COBALT framework for categorical structural optimization under algebraic uncertainty, three benchmark beam structures and two high dimensional test problems. The 10
Algorithm 1 COBALT for categorical optimization 1: Input: Categorical catalog X (n instances), # of variables e, evaluations T , parameter κ 2: Output: Robust best valid categorical design Z∗T 3: // Discrete Manifold Anchoring 4: Normalize attributes of X column-wise within [0, 1] 5: Apply Isomap Φ mapping X to latent representation Y 6: Lock mapped coordinates as discrete anchor set Acat = {y1 , . . . , yn } ⊂ Rm
7: Define feasible search domain as discrete tensor grid ΩD ≡ (Acat )e 8: Evaluate initial random designs via MC-FEA to initialize dataset D0 9: Identify incumbent best design Z∗0 , initialize trust-region length L1 10: for t = 1, . . . , T do 11: // Random Decomposition 12: Sample uniform random spanning tree graph Gt = (V, Et ) 13: 14: 15: 16: 17: 18: 19: 20: 21:
// Trust-Region Discrete Graph Acquisition Train Additive SAAS-GP surrogate on Dt−1 using tree-decomposed kernel kGt Marginalize hyperparameters via MCMC for predictive mean µt (Z) and variance σ2t (Z) Construct additive LCB acquisition function factorized over Et : P αLCB (Z) = (u,v)∈Et µuv (yu , yv ) − κ · σuv (yu , yv ) Define dynamically scaled hyper-cubic Trust Region T Rt centered at Z∗t−1 Define constrained search space intersection Ω(t) T R = Ω D ∩ T Rt Discrete Search: Use evolutionary operators traversing Acat to purely optimize αLCB : Znext = arg min αLCB (Z) Z∈Ω(t) TR
// Stochastic MC-FEA Oracle Physically evaluate exact discrete structure Znext via MC-FEA under uncertainty ξ (t) Obtain heteroscedastic noisy robust observation y(t) obs = Jrobust (Znext ) + ϵ (t) 25: Augment dataset: Dt = Dt−1 ∪ {(Znext , yobs )} 26: Update robust incumbent best design Z∗t 27: Expand or contract trust-region length Lt+1 based on recent performance 28: end for 29: Return Optimal robust structurally valid design Z∗T
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detailed information of the five numerical tests is listed in Tab. 1, where the first three benchmark tests are carried out to evaluate the validity of the proposed method on planar and spatial structures, and the last two examples are tested to illustrate the capability of the proposed method in dealing with medium and large scale variables with uncertainties. of increasing complexity are investigated: a planar ten-beam truss and a spatial 120-beam dome. For each example, the robust objective Jrobust and robust constraints Grobust defined in Eq. (1) are evaluated via Monte Carlo-based Finite Element Analysis (MC-FEA), where the uncertain parameters ξ encompass material property fluctuations and stochastic external loads. All experiments are conducted on a standard desktop workstation. Table 1: The information of the five numerical examples.
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Validity benchmark test 3D benchmark test 3D benchmark test Medium scale variables large scale of variables
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Figure 3: The load-case illustration of the ten-beam structure.
• Continuous Relaxation BO (CR-BO): A standard Latent Space Bayesian Optimization approach that fits continuous polynomial interpolations to the reduced manifold, optimizes a continuous acquisition function via gradient-based methods, and applies nearest-neighbor rounding-off to recover discrete categorical instances. • Heuristic Genetic Algorithm (GA): A conventional genetic algorithm operating directly on the categorical encoding without dimensionality reduction or surrogate assistance. • Random Search (RS): Uniform random sampling over the full combinatorial space ΩD , serving as a lowerbound reference. For all BO-based methods, the SAAS-GP surrogate described in Section 3.2 is employed with identical hyperprior settings to ensure a fair comparison. The key differentiator is the acquisition maximization strategy: COBALT exclusively traverses the discrete anchored graph, while CR-BO relies on continuous relaxation followed by rounding-off. 4.1. The planar ten-beam truss The first benchmark is the classical ten-beam planar truss. The truss consists of 10 beam members connected at 6 nodes, as illustrated in Fig. 3. The 10 beams are divided into 4 groups: the horizontal group, the vertical group, the sub-diagonal group, and the principal diagonal group. Each group shares the same cross-section. The left two nodes are fixed on a rigid wall and the lower-right node bears a downward load of 10000 N. Each beam member is assigned a categorical cross-section variable xi (i = 1, . . . , 10) selected from a predefined catalog of n standard steel profiles. The catalog instances differ simultaneously in cross-sectional area A and the moments of inertia Iy and Iz , forming a multi-dimensional attribute representation. The material density is ρ = 7850 kg/m3 and the gravitational acceleration is g = 9.81 m/s2 The robust optimization problem is formulated as: r 1 T 1 min.: J(x1 , x2 , x3 , x4 ) = Eξ [ u Ku] + γ Vξ [ uT Ku]; 2 2 s. t.: m(x1 , x2 , x3 , x4 ) − 240 ≤ 0; Ku = P; max(Fcry − F) ≤ 0, Fcry = ( fycr1 , fycr2 , · · · , fycr10 );
(19)
max(Fcrz − F) ≤ 0, Fcrz = ( fzcr1 , fzcr2 , · · · , fzcr10 ); xi ∈ {x1i , x2i , · · · , x54 i }, i = 1, 2, 3, 4; xji = (Aji , Iy ji , Iz ji )T , j = 1, 2, · · · , 54. where J is the objective function combining the expectation and standard deviation of the structural strain energy. γ is the weight factor on the standard deviation, set to 1 in this study. u denotes the structural displacement vector, and 21 uT Ku is the structural strain energy. xi denotes the i-th design variable. Ku = P represents the finite element governing equation. Fcry and Fcrz are the critical buckling load vectors in the y- and z-directions, respectively. The uncertain parameter vector ξ includes: 12
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• Young’s modulus: E ∼ N(E0 , (0.05E0 )2 ), representing a 5% coefficient of variation due to material manufacturing tolerances. E0 = 2.1e11Pa in this work. • External loads: Pi ∼ N(Pi,0 , (0.10Pi,0 )2 ), representing a 10% coefficient of variation due to environmental load fluctuations. Pi,0 = 1000N in this work. For each candidate design evaluated during the optimization, the robust metrics are estimated via MC-FEA with N MC = 500 Monte Carlo samples, introducing heteroscedastic observation noise as modeled in Eq. (2). The Isomap-based dimensionality reduction maps the M-dimensional attribute vectors of each categorical variable to an m-dimensional latent representation (m ≪ M), and the resulting discrete anchor set Acat (Eq. (8)) forms the foundation of the COBALT search space. The total latent dimensionality is D = m × 10. Fig. 4 presents the convergence histories of the robust objective Jrobust , the buckling constraints in y and z directions and the mass constraint for all iterations over 200 MC-FEA evaluations. After 40 iterations, the robust objective converges and all the design constraints are satisfied. The COBALT framework achieves the lowest robust objective with the fastest convergence rate. Notably, CR-BO initially converges at a comparable pace but stagnates prematurely due to repeated decoding failures: the continuous acquisition optimizer frequently identifies pseudo-optima in the latent continuum that, upon nearest-neighbor rounding-off, map to structurally inferior or constraint-violating configurations. In contrast, COBALT’s discrete graph acquisition guarantees that every recommended configuration is physically admissible, enabling monotonic improvement throughout the optimization campaign. Fig. 5 compares the latent-space distributions obtained by different manifold learning methods for the ten-beam optimization problem. The upper-row plots show the embedded catalog instances colored by cross-sectional area, while the lower-row plots report the corresponding failure probability and constraint-violation tendency over the same latent coordinates. The comparison indicates that the quality of the latent representation strongly affects the physical consistency of the categorical search space. Methods that better preserve the topology of the original attribute space produce smoother area gradients and more coherent feasible regions, whereas distorted embeddings scatter physically dissimilar sections into neighboring latent positions and lead to irregular failure-probability patterns. These results confirm that an appropriate manifold representation is essential for constructing reliable anchored latent graphs and reducing decoding-related constraint violations. Fig. 6 further shows that deterministic manifold-optimization strategies become fragile when uncertainty is introduced. As the coefficient of variation increases, the failure probability rises and constraint violations remain methoddependent, indicating that deterministic latent representations alone cannot reliably handle categorical optimization under uncertainty. Fig. 7 illustrates the optimization trajectory of COBALT in the original physical attribute space, where each design variable is represented by its cross-sectional attributes (A, Iy , Iz ). The search path progresses from the initial anchor (solid square) toward the best feasible design (star marker) through a sequence of catalog-valid discrete steps, demonstrating the structured exploration of the combinatorial design space. Fig. 8 further visualizes the corresponding 13
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optimization trajectory in the low-dimensional latent space under uncertainty. The pink circles represent the discrete catalog anchors embedded in the latent space, while the blue squares indicate successive BO-evaluated steps. The solid blue path traces the progression from the initial anchor (orange square) to the global best feasible design (red star), and the dashed segment records the exploratory steps taken after the current best was identified. Every evaluated candidate coincides with a catalog anchor, confirming that the graph-based search operates exclusively on physically valid discrete instances throughout the optimization. 4.2. The spatial dome structure The second benchmark is a large-scale 120-beam spatial dome structure, which poses a significantly more challenging categorical optimization problem due to its higher combinatorial complexity and richer structural behavior under uncertainty. As shown in Fig. 9, the 120 beams are partitioned into seven symmetry-preserving design groups, yielding seven categorical design variables. Each group shares a common cross-section variable, and all candidate sections are selected from the same standard steel profile catalog used in the ten-beam example. The robust optimization problem is formulated analogously to Eq. (19), with additional consideration of local buckling constraints: q min.: J(x1 , x2 , · · · , x7 ) = Eξ [0.5uT Ku] + γ Vξ [0.5uT Ku]; s. t.:
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(20)
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The MC-FEA uses N MC = 500 samples per evaluation. The total combinatorial search space is ne , which is astronomically large for the 120-beam dome. Fig. 10 presents the convergence histories of the robust objective, the mass constraint, and the buckling constraints over 200 MC-FEA evaluations for the 120-beam dome. The performance gap between COBALT and the baseline methods is substantially amplified compared to the ten-beam case, reflecting the increased difficulty of navigating a higher-dimensional combinatorial space under uncertainty. COBALT achieves the lowest robust objective with the fastest convergence rate among all compared methods. The GA and RS methods fail to locate competitive feasible designs within the allocated evaluation budget, confirming the intractability of brute-force approaches in high-dimensional categorical OUU. CR-BO demonstrates reasonable initial progress but stagnates prematurely due to an escalating decoding failure rate: as the number of categorical variables increases, the probability of a continuous pseudo-optimum mapping to a topologically distant discrete anchor grows combinatorially, leading to structurally inferior or constraint-violating configurations. In contrast, COBALT’s discrete graph acquisition guarantees that every recommended configuration is physically admissible, enabling monotonic improvement throughout the optimization campaign. Fig. 11 illustrates the optimization trajectory of COBALT in the original physical attribute space, where each of the seven design variables is represented by its cross-sectional attributes (A, Iy , Iz , J x ). The search path progresses from 15
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Figure 7: The optimization path in the original physical attribute space for the ten-beam optimization problem.
the initial anchor toward the best feasible design through a sequence of catalog-valid discrete steps, demonstrating the structured exploration of the seven-dimensional combinatorial design space. Fig. 12 further visualizes the corresponding optimization trajectory in the low-dimensional latent space under uncertainty. The discrete catalog anchors are embedded as fixed points in the latent manifold, and the blue squares indicate successive BO-evaluated steps. The solid path traces the progression from the initial anchor (orange square) to the global best feasible design (red star), and the dashed segment records the exploratory steps taken after the current best was identified. Every evaluated candidate coincides with a catalog anchor, confirming that the graph-based search operates exclusively on physically valid discrete instances throughout the optimization. 4.3. The spatial six-story frame structure The sixe-story rigid frame structure contains 63 beams which are divided into eight groups (Fig. 13). As previous, the beams in the same group will be assigned the same cross-section chosen from 54 cross-section instances, as previous, together with their physical attributes: areas, area moments of inertia about y-axis, area moments of inertia about z-axis and torsion moments of inertia about x-axis. The rigid frame structure bears three kinds of loads: 1) The gravity load on the floor (19.16kPa); 2) The dead load of the beams; 3) The lateral load caused by the wind (110kN). This six-story rigid frame example poses a significantly more challenging categorical optimization problem due to its higher combinatorial complexity and richer structural behavior under uncertainty. The robust optimization problem is formulated analogously to Eq. (19), with additional consideration of local
16
Design Variable 1
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Figure 8: The optimization path in low dimensional design space with uncertainties for the ten-beam optimization problem.
buckling constraints: min.:
J(x1 , x2 , · · · , x7 ) = Eξ [0.5uT Ku] + γ
s. t.:
m(x1 , x2 , · · · , x8 ) − 5000 ≤ 0;
q Vξ [0.5uT Ku];
Ku = P; max(Fcry − F) ≤ 0, Fcry = ( fycr1 , fycr2 , · · · , fycr63 );
(21)
max(Fcrz − F) ≤ 0, Fcrz = ( fzcr1 , fzcr2 , · · · , fzcr63 ); xi ∈ {x1i , x2i , · · · , x54 i }, i = 1, 2, · · · , 8; xji = (Aji , Iy ji , Iz ji , J x ji )T , j = 1, 2, · · · , 54. where U denotes the total strain energy (a measure of structural compliance), F cr j is the critical Euler buckling load act of beam j, F j is the actual compressive force, and γ1 , γ2 , βbuck are the respective penalty and reliability parameters. The inclusion of strain energy in the objective promotes designs that are not only lightweight but also globally stiff under uncertain loading conditions. The uncertainty model is extended to include: • Young’s modulus: E ∼ N(E0 , (0.05E0 )2 ). • External loads: Pi ∼ N(Pi,0 , (0.10Pi,0 )2 ). 17
Figure 9: The load-case illustration of the dome structure. 0
18.5
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Buck-Y Buck-Z Mass
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1 Mass constraint (Kg)
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Figure 10: The convergence history of the dome optimization: (a) robust objective convergence, (b) mass and buckling-constraint evolution, and (c) feasible anchored designs in the latent space.
• Geometric imperfections: nodal coordinate perturbations ∆rk ∼ N(0, (0.001L)2 ), where L is the characteristic span length. The MC-FEA uses N MC = 500 samples per evaluation. The total combinatorial search space is ne , which is astronomically large for the eight-variable six-story frame. COBALT achieves the lowest robust objective with the fastest convergence rate among all compared methods. The GA and RS methods fail to locate competitive feasible designs within the allocated evaluation budget, confirming the intractability of brute-force approaches in high-dimensional categorical OUU. CR-BO demonstrates reasonable initial progress but stagnates prematurely due to an escalating decoding failure rate: as the number of categorical variables increases, the probability of a continuous pseudo-optimum mapping to a topologically distant discrete anchor grows combinatorially, leading to structurally inferior or constraint-violating configurations. In contrast, COBALT’s discrete graph acquisition guarantees that every recommended configuration is physically admissible, enabling monotonic improvement throughout the optimization campaign. Fig. 14 illustrates the optimization trajectory of COBALT in the original physical attribute space, where each of the eight design variables is represented by its cross-sectional attributes (A, Iy , Iz , J x ). The search path progresses from the initial anchor toward the best feasible design through a sequence of catalog-valid discrete steps, demonstrating the structured exploration of the eight-dimensional combinatorial design space. Fig. 15 further visualizes the corresponding optimization trajectory in the low-dimensional latent space under uncertainty. The discrete catalog anchors are embedded as fixed points in the latent manifold, and the blue squares indicate successive BO-evaluated steps. The solid path traces the progression from the initial anchor (orange square) to the global best feasible design (red star), and the dashed segment records the exploratory steps taken after the current 18
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Sections (catalog) COBALT step Start (iter -9) Start → best (solid) After best (dashed) Global best feasible
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Figure 11: The optimization path in the original physical attribute space for the dome optimization problem. Design Variable 2
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Figure 12: The optimization path in low dimensional design space with uncertainties for the dome optimization problem.
best was identified. Every evaluated candidate coincides with a catalog anchor, confirming that the graph-based search operates exclusively on physically valid discrete instances throughout the optimization. 4.4. The planer 105-beam structure The fourth benchmark is a medium-scale planar 105-beam truss structure. The 105 beams are treated as independent categorical design variables, each selected from the same catalog of 54 standard steel profiles with attributes (A, Iy , Iz ). This example is specifically designed to evaluate COBALT’s capability in handling a medium-scale combinatorial space under uncertainty, where the number of design variables substantially exceeds those in the preceding benchmarks.
19
4 1
3 5
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Figure 13: The load-case illustration of the six-story frame structure.
The robust optimization problem is formulated as: min.:
J(x1 , x2 , x3 , x4 ) = Eξ [0.5uT Ku] + γ
s. t.:
m(x1 , x2 , · · · , x105 ) − 5000 ≤ 0;
q Vξ [0.5uT Ku];
Ku = P; max(Fcry − F) ≤ 0, Fcry = ( fycr1 , fycr2 , · · · , fycr105 );
(22)
max(Fcrz − F) ≤ 0, Fcrz = ( fzcr1 , fzcr2 , · · · , fzcr105 ); xi ∈ {x1i , x2i , · · · , x54 i }, i = 1, 2, · · · , 105; xji = (Aji , Iy ji , Iz ji )T , j = 1, 2, · · · , 54. COBALT achieves the lowest robust objective with the fastest convergence rate among all compared methods. The GA and RS methods fail to locate competitive feasible designs within the allocated evaluation budget, confirming the intractability of brute-force approaches in high-dimensional categorical OUU. CR-BO demonstrates reasonable initial progress but stagnates prematurely due to an escalating decoding failure rate: with 105 independent categorical variables, the probability of a continuous pseudo-optimum mapping to a topologically distant discrete anchor grows substantially, leading to structurally inferior or constraint-violating configurations. In contrast, COBALT’s discrete graph acquisition guarantees that every recommended configuration is physically admissible, enabling monotonic improvement throughout the optimization campaign. Fig. 17 illustrates the optimization trajectory of COBALT in the original physical attribute space, where each of the 105 design variables is represented by its cross-sectional attributes (A, Iy , Iz ). The search path progresses from the initial anchor toward the best feasible design through a sequence of catalog-valid discrete steps, demonstrating COBALT’s ability to navigate a large-scale combinatorial design space in a structured and physically consistent manner. 20
Design Variable 1
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Figure 14: The optimization path in the original physical attribute space for the six-story frame optimization problem. Design Variable 2
Design Variable 1
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Figure 15: The optimization path in low dimensional design space with uncertainties for the six-story frame optimization problem.
Fig. 18 further visualizes the corresponding optimization trajectory in the low-dimensional latent space under uncertainty. The discrete catalog anchors are embedded as fixed points in the latent manifold, and the blue squares indicate successive BO-evaluated steps. The solid path traces the progression from the initial anchor (orange square) to the global best feasible design (red star), and the dashed segment records the exploratory steps taken after the current best was identified. Every evaluated candidate coincides with a catalog anchor, confirming that the graph-based search operates exclusively on physically valid discrete instances throughout the optimization. 4.5. The high dimensional 1564-beam structure The fifth benchmark is a high-dimensional spatial 1564-beam structure, representing the most challenging categorical optimization problem considered in this study. The 1564 beams are treated as independent categorical design variables, each selected from the same catalog of 54 standard steel profiles with attributes (A, Iy , Iz , J x ). This example is specifically designed to evaluate COBALT’s scalability in handling an extremely large combinatorial space under uncertainty.
21
Figure 16: The load-case illustration of the 105-beam structure.
The robust optimization problem is formulated as: min.:
J(x1 , x2 , · · · , x7 ) = Eξ [0.5uT Ku] + γ
s. t.:
m(x1 , x2 , · · · , x8 ) − 70000 ≤ 0;
q Vξ [0.5uT Ku];
Ku = P; max(Fcry − F) ≤ 0, Fcry = ( fycr1 , fycr2 , · · · , fycr1564 ); max(Fz − F) ≤ 0, Fz = ( fz , fz , · · · , fz cr
cr
cr1
cr2
cr1564
(23)
);
xi ∈ {x1i , x2i , · · · , x54 i }, i = 1, 2, · · · , 1564; xji = (Aji , Iy ji , Iz ji , J x ji )T , j = 1, 2, · · · , 54. COBALT achieves the lowest robust objective with the fastest convergence rate among all compared methods. The GA and RS methods fail to locate competitive feasible designs within the allocated evaluation budget, confirming the complete intractability of brute-force approaches at this scale. CR-BO demonstrates reasonable initial progress but stagnates prematurely due to an escalating decoding failure rate: with 1564 independent categorical variables, the probability of a continuous pseudo-optimum mapping to a topologically distant discrete anchor becomes prohibitively large, leading to structurally inferior or constraint-violating configurations. In contrast, COBALT’s discrete graph acquisition guarantees that every recommended configuration is physically admissible, and the SAAS-GP’s automatic dimensionality shielding effectively identifies the most influential latent features, enabling efficient convergence even at this extreme scale. Fig. 20 illustrates the optimization trajectory of COBALT in the original physical attribute space, where each of the 1564 design variables is represented by its cross-sectional attributes (A, Iy , Iz , J x ). The search path progresses from the initial anchor toward the best feasible design through a sequence of catalog-valid discrete steps, demonstrating COBALT’s ability to navigate an extremely large-scale combinatorial design space in a structured and physically consistent manner. Fig. 21 further visualizes the corresponding optimization trajectory in the low-dimensional latent space under uncertainty. The discrete catalog anchors are embedded as fixed points in the latent manifold, and the blue squares indicate successive BO-evaluated steps. The solid path traces the progression from the initial anchor (orange square) to the global best feasible design (red star), and the dashed segment records the exploratory steps taken after the current best was identified. Every evaluated candidate coincides with a catalog anchor, confirming that the graph-based search operates exclusively on physically valid discrete instances throughout the optimization.
22
Design Variable 1
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Sections COBALT step Start Start best (solid) After best (dashed) Best feasible
4)
Iy (m
Figure 17: The optimization path in the original physical attribute space for the 105-beam optimization problem.
4.6. Parameter sensitivity study To investigate the influence of key algorithmic parameters on the COBALT framework’s performance, we conduct a systematic sensitivity analysis using the ten-beam truss as the testbed. 4.6.1. Effect of trust-region scaling The trust-region radius directly controls the balance between local exploitation and global exploration in the discrete graph acquisition. The convergence behavior under different initial trust-region scaling factors indicates that an overly conservative trust region leads to slow convergence due to insufficient exploration, while an excessively aggressive trust region degrades performance by diluting the exploitation of promising regions. The adaptive scaling mechanism described in Section 3.3 effectively mitigates this sensitivity by dynamically adjusting the trust-region boundary based on the optimization progress. 4.6.2. Effect of Monte Carlo sample size The number of MC-FEA samples N MC per evaluation governs the trade-off between observation noise level and computational cost. Under varying N MC , fewer samples (N MC = 100) amplify the heteroscedastic noise, causing the SAAS-GP surrogate to require more evaluations to achieve reliable predictions. Conversely, larger sample sizes (N MC = 1000) reduce noise but proportionally increase the per-evaluation computational cost. The default setting of N MC = 500 provides a practical balance, and the SAAS-GP’s fully Bayesian noise modeling (Eq. (12)) demonstrates robustness across the tested range. 4.6.3. Effect of SAAS sparsity prior The SAAS prior’s half-Cauchy scale parameter controls the degree of automatic dimensionality shielding. Comparisons under different sparsity strengths show that a stronger sparsity prior aggressively suppresses irrelevant latent dimensions, which is beneficial when the effective dimensionality is low but may discard useful information in more 23
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Figure 18: The optimization path in low dimensional design space with uncertainties for the 105-beam optimization problem.
complex problems. The default setting achieves robust performance across both the ten-beam and 120-beam examples. 5. Conclusions This paper has presented the COBALT (Categorical Optimization via Bayesian Anchored Latent Trust-Regions) framework, a principled methodology for robust categorical structural optimization under high-dimensional aleatoric uncertainty. The framework addresses two fundamental bottlenecks that have long plagued existing approaches: the decoding failure induced by continuous relaxation and rounding-off, and the computational intractability of uncertainty-aware evaluation over massive combinatorial spaces. The key contributions and conclusions are summarized as follows. First, by mathematically locking the dimensionalityreduced latent representation as an absolute discrete anchor set, COBALT permanently severs the root cause of decoding failures. Any continuous relaxation of the categorical manifold is strictly forbidden. The global feasible search domain is rigorously confined to a discrete Cartesian product tensor grid, guaranteeing that every candidate configuration recommended by the optimizer corresponds to a physically valid catalog instance. Across all numerical experiments, COBALT eliminates decoding failures by construction, in stark contrast to continuous relaxation-based Bayesian optimization, whose failure rate escalates with problem dimensionality. Second, the integration of Sparse Axis-Aligned Subspace (SAAS) priors into the Gaussian Process surrogate enables uncertainty-aware sparse inference under the heteroscedastic noise inherent to Monte Carlo-based Finite Element Analysis. The fully Bayesian treatment of hyperparameters allows the surrogate to automatically identify and shield irrelevant latent dimensions, maintaining high predictive fidelity even when the number of categorical variables is large and the observation budget is severely limited. Third, the graph-based evolutionary operators, powered by Dijkstra’s shortest-path traversal of the discrete anchored network, serve as a zero-cost internal acquisition maximizer within dynamically scaled latent trust regions. By replacing continuous gradient-based acquisition optimizers entirely, COBALT eliminates the need for any rounding-off post-processing. The computational overhead of the discrete graph acquisition is negligible relative to the dominant MC-FEA cost, confirming the practical efficiency of the proposed search architecture. Numerical validation across the ten-beam planar truss, the 120-beam spatial dome, the spatial six-story frame, the planar 105-beam structure, and the high-dimensional 1564-beam structure demonstrates that COBALT consistently outperforms conventional heuristic genetic algorithms, random search, and continuous relaxation Bayesian optimization in terms of robust objective quality, convergence speed, and physical admissibility. The performance advantage becomes more pronounced as the scale and dimensionality of the categorical design problem increase, underscoring the scalability of the framework. Several directions merit further investigation. Extending COBALT to mixed-variable 24
Figure 19: The load-case illustration of the high dimensional 1564-beam structure.
problems involving both categorical and continuous design variables would broaden its applicability to a wider class of engineering design tasks. Incorporating multi-fidelity MC-FEA evaluations could further reduce the computational budget required for uncertainty quantification. Additionally, exploring alternative manifold learning techniques beyond Isomap, such as Bayesian Gaussian Process Latent Variable Models, may yield richer latent representations for highly non-linear categorical catalogs. These extensions are left for future work. Acknowledgment The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (12202157). We also express our sincere thanks to the Exploration Foundation of the Key Laboratory of CNC Equipment Reliability, Ministry of Education and the National Key Laboratory of Automotive Chassis Integration and Bionics, School of Mechanical and Aerospace Engineering, Jilin University. References [1] R. F. Coelho, M. Xiao, A. Guglielmetti, M. Herrera, W. Zhang, Investigation of three genotypes for mixed variable evolutionary optimization (2015) 309–319. [2] M. Kokkolaras, C. Audet, J. E. Dennis, Mixed variable optimization of the number and composition of heat intercepts in a thermal insulation system, Optimization and Engineering 2 (2001) 5–29. [3] P. Lindroth, M. Patriksson, Pure Categorical Optimization: A Global Descent Approach, Department of Mathematical Sciences, Chalmers University of Technology, University of Gothenburg, 2011. [4] D. Sloane, S. P. Morgan, An introduction to categorical data analysis, Annual review of sociology (1996) 351–375. [5] F. Herrera, M. Lozano, J. L. Verdegay, Tackling real-coded genetic algorithms: Operators and tools for behavioural analysis, Artificial intelligence review 12 (1998) 265–319. [6] D. E. Goldberg, Genetic algorithms, Pearson Education India, 2006. [7] R. A. Caruana, J. D. Schaffer, Representation and hidden bias: Gray vs. binary coding for genetic algorithms, in: Machine Learning Proceedings 1988, Elsevier, 1988, pp. 153–161. 25
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Figure 20: The optimization path in the original physical attribute space for the 1564-beam optimization problem.
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Figure 21: The optimization path in low dimensional design space with uncertainties for the 1564-beam optimization problem.
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