CASMOPOLITAN: High-Dimensional Mixed-Space BO Optimizer

• CASMOPOLITAN is a high-dimensional Bayesian optimization method for categorical and mixed-variable spaces that employs separate continuous and categorical trust regions.
• It leverages a Transformed Overlapped kernel with ordinal encoding to capture complex relationships between continuous and discrete variables.
• Integration into MOCA-HESP demonstrates enhanced performance on benchmark problems, validating its dual-region local search strategy.

Searching arXiv for CASMOPOLITAN and related Bayesian optimization papers to ground the article with current paper metadata and citations. CASMOPOLITAN is a high-dimensional Bayesian optimization method for categorical and mixed spaces whose defining mechanism, as summarized in later work, is a local search or trust-region strategy that maintains separate continuous and categorical trust regions around the best solution found so far (Ngo et al., 9 Aug 2025). In the framing of "MOCA-HESP: Meta High-dimensional Bayesian Optimization for Combinatorial and Mixed Spaces via Hyper-ellipsoid Partitioning" (Ngo et al., 9 Aug 2025), CASMOPOLITAN is both a state-of-the-art optimizer in its own right and a base optimizer that can be wrapped by a meta-algorithm, yielding the derived method $\textbf{MOCA-HESP-Casmo}$. The same source presents CASMOPOLITAN as a strong method for high-dimensional combinatorial and mixed-variable Bayesian optimization rather than as a continuous-only BO procedure, and it treats ordinal variables as categorical in that setting (Ngo et al., 9 Aug 2025).

1. Problem domain and methodological position

CASMOPOLITAN is described as “a state-of-the-art high-dimensional BO method for categorical and mixed spaces” and is grouped with methods for high-dimensional combinatorial and mixed spaces (Ngo et al., 9 Aug 2025). In the problem formulation used by the later meta-framework, the search space is written as $\mathcal{Z}=\mathcal{X}\times\mathcal{H}$, where $\mathcal{X}$ is continuous and $\mathcal{H}$ is combinatorial, and the optimization objective is

$$\bm{z}^* \in \arg\min_{\bm{z}\in\mathcal{Z}} f(\bm{z}).$$

Within that framing, CASMOPOLITAN is intended for high-dimensional categorical, combinatorial, and mixed-variable regimes rather than for low-dimensional continuous domains alone (Ngo et al., 9 Aug 2025).

A notable feature of the presentation in (Ngo et al., 9 Aug 2025) is that CASMOPOLITAN is not treated merely as a baseline. It is explicitly incorporated into a higher-level method as one of three base Bayesian optimization optimizers. The resulting wrapped method is named $\textbf{MOCA-HESP-Casmo}$, alongside $\textbf{MOCA-HESP-BO}$ and $\textbf{MOCA-HESP-Bounce}$ (Ngo et al., 9 Aug 2025). This positioning is significant because it characterizes CASMOPOLITAN as sufficiently mature and modular to serve as a substrate for meta-level augmentation.

2. Core local-region mechanism

The central mechanism attributed to CASMOPOLITAN is a trust-region scheme designed to make high-dimensional search tractable (Ngo et al., 9 Aug 2025). According to the summary in Appendix $\ref{sec:appendix_casmo}$ of that paper, CASMOPOLITAN maintains two separate trust regions.

The continuous trust region is “a hyper-rectangle centered at the best solution found so far, with its side lengths determined by the GP length-scales multiplied with a length ratio factor $L_x$” (Ngo et al., 9 Aug 2025). The categorical trust region is “constructed as a region centered at the best solution found so far, and contain the data points within a Hamming distance of $\mathcal{Z}=\mathcal{X}\times\mathcal{H}$0 to the TR center” (Ngo et al., 9 Aug 2025). During optimization, the method “adaptively expands or shrinks the TRs (increases or decreases both $\mathcal{Z}=\mathcal{X}\times\mathcal{H}$1 and $\mathcal{Z}=\mathcal{X}\times\mathcal{H}$2) depending on the success or failure of the algorithm” (Ngo et al., 9 Aug 2025).

This dual-region design is the distinctive operational signature of CASMOPOLITAN in the provided literature. The method does not collapse mixed-variable locality into a single continuous geometry. Instead, it preserves a continuous neighborhood structure for real-valued variables and a Hamming-ball structure for categorical variables. A plausible implication is that CASMOPOLITAN encodes locality in a type-aware way: Euclidean-style locality for continuous coordinates and discrete locality for categorical assignments.

3. Surrogate modeling, encoding, and acquisition

The surrogate-modeling component emphasized in (Ngo et al., 9 Aug 2025) is the Transformed Overlapped kernel. In the related-work summary, CASMOPOLITAN is said to “improv[e] the expressiveness of the model by learning different GP lengthscales for categorical kernels,” and the appendix states that the kernel “can capture the relationship between categorical variables, and also relationship between categorical and continuous variables” (Ngo et al., 9 Aug 2025). This places the method within GP-based BO, but with a kernel explicitly tailored to noncontinuous structure.

Because this kernel uses Hamming distance, CASMOPOLITAN “simply uses ordinal encoding to transform categorical data into numerical data for GP modelling” (Ngo et al., 9 Aug 2025). In the same paper’s general framing, ordinal variables are treated as categorical. The resulting design is therefore not an arbitrary encoding-plus-kernel combination; it is a matched pair in which ordinal encoding supplies numerical representations while the kernel and trust-region logic continue to operate with categorical semantics.

The acquisition mechanism preserved in the meta-framework is TS. Appendix baseline details state: “For the acquisition function, we use TS” (Ngo et al., 9 Aug 2025). The same appendix reports the following baseline settings for CASMOPOLITAN: $\mathcal{Z}=\mathcal{X}\times\mathcal{H}$3, $\mathcal{Z}=\mathcal{X}\times\mathcal{H}$4, $\mathcal{Z}=\mathcal{X}\times\mathcal{H}$5, $\mathcal{Z}=\mathcal{X}\times\mathcal{H}$6, $\mathcal{Z}=\mathcal{X}\times\mathcal{H}$7, and $\mathcal{Z}=\mathcal{X}\times\mathcal{H}$8 (Ngo et al., 9 Aug 2025). These details indicate that the method is operationalized as a concrete trust-region BO procedure rather than an abstract heuristic.

4. Integration into MOCA-HESP

The most detailed recent account of CASMOPOLITAN in the provided material comes from its incorporation into MOCA-HESP (Ngo et al., 9 Aug 2025). That paper states that “incorporating MOCA-HESP with CASMOPOLITAN … is especially challenging due to their distinct core features, such as the different local region strategies … as well as their different requirements such as input data encoding” (Ngo et al., 9 Aug 2025). The challenge arises because CASMOPOLITAN already has a nontrivial local-region mechanism, whereas MOCA-HESP introduces hyper-ellipsoid space partitioning in an encoded continuous space.

The HESP side defines a Gaussian search distribution

$\mathcal{Z}=\mathcal{X}\times\mathcal{H}$9

and a base $\mathcal{X}$0-confidence hyper-ellipsoid

$\mathcal{X}$1

where the Mahalanobis distance is given in the paper as

$\mathcal{X}$2

For combinatorial and mixed spaces, MOCA-HESP first introduces an encoder $\mathcal{X}$3, optimizes in $\mathcal{X}$4, and decodes back with $\mathcal{X}$5 via nearest-code decoding (Ngo et al., 9 Aug 2025).

The CASMOPOLITAN-specific integration is the definition of a new local region that must satisfy both a Mahalanobis-distance criterion and a Hamming-distance criterion:

$\mathcal{X}$6

The scaling vector is defined dimensionwise by $\mathcal{X}$7 for continuous dimensions and $\mathcal{X}$8 otherwise, with covariance transformed via $\mathcal{X}$9 (Ngo et al., 9 Aug 2025).

Aspect	CASMOPOLITAN	MOCA-HESP-Casmo
Continuous locality	Hyper-rectangle controlled by $\mathcal{H}$0	Anisotropically scaled HESP ellipsoid
Categorical locality	Hamming-distance trust region with threshold $\mathcal{H}$1	Same Hamming constraint retained
GP and acquisition	Ordinal encoding, Transformed Overlapped kernel, TS	Same surrogate family and acquisition preserved

The paper is explicit that MOCA-HESP does not replace CASMOPOLITAN’s BO core. It preserves “the kernel and input encoding type” when fitting the GP and uses “the same acquisition function employed by the BO optimizer” (Ngo et al., 9 Aug 2025). Accordingly, the augmentation occurs primarily in local-region geometry and partitioning.

5. Empirical role and observed performance

The empirical role of CASMOPOLITAN in (Ngo et al., 9 Aug 2025) is twofold: it appears separately as a baseline, and it also appears as the embedded optimizer inside $\mathcal{H}$2. The benchmark suite contains 9 problems: Ackley20c, Shifted Ackley20c, Ackley53m, Antibody Design, LABS, MaxSAT28, MaxSAT125, Shifted LABS, and Cellular Network, spanning dimensions from 11 to 125 and category cardinalities from 2 to 20 (Ngo et al., 9 Aug 2025). The comparison set includes CASMOPOLITAN, the other MOCA-HESP-derived methods and their base optimizers, TPE, SMAC, CMA-ES, and Random Search (Ngo et al., 9 Aug 2025).

The key result reported for the CASMOPOLITAN-derived variant is that “Compared to CASMOPOLITAN, MOCA-HESP-Casmo shows superior performance on 7 out of the 9 benchmark problems,” and that in those 7 problems it “consistently outperforms CASMOPOLITAN across all iterations” (Ngo et al., 9 Aug 2025). The figure caption reinforces the same conclusion: “MOCA-HESP-Casmo outperforms CASMOPOLITAN” (Ngo et al., 9 Aug 2025). Within the bounds of the provided evidence, the main lesson is not that CASMOPOLITAN is weak, but that its local BO machinery can benefit from being placed inside a hyper-ellipsoid partitioning framework.

Runtime data indicate moderate overhead for the wrapped variant. Average per-iteration examples reported in Table $\mathcal{H}$3 include Ackley20c: 4.82s for CASMOPOLITAN versus 6.15s for MOCA-HESP-Casmo; Antibody Design: 6.33s versus 10.82s; MaxSAT125: 11.78s versus 19.65s; Ackley53m: 18.07s versus 18.17s; and Cellular Network: 7.58s versus 8.40s (Ngo et al., 9 Aug 2025). The paper characterizes HESP as incurring “minimal computation overhead” in general, while the tabulated values suggest that the additional cost for the CASMOPOLITAN-based variant is benchmark-dependent.

6. Interpretation, scope, and related conceptual usage

The later literature represented here treats CASMOPOLITAN as a strong base optimizer rather than as a method requiring wholesale replacement (Ngo et al., 9 Aug 2025). Its reported strengths are clear: it is state-of-the-art for high-dimensional categorical and mixed BO, it uses local regions to make high-dimensional search tractable, and it employs an expressive transformed overlapped kernel for categorical and mixed relations (Ngo et al., 9 Aug 2025). The paper does not present a long critique of CASMOPOLITAN itself. Instead, it identifies the integration of its separate continuous and categorical trust regions into HESP as “especially challenging,” which suggests that the method’s locality mechanism is specialized and structurally important (Ngo et al., 9 Aug 2025).

A separate 2025 paper on urban human mobility simulation, "CAMS: A CityGPT-Powered Agentic Framework for Urban Human Mobility Simulation" (Du et al., 16 Jun 2025), uses the phrase “quite close in spirit to a CASMOPOLITAN-style simulator” when describing a system that balances individual-level behavior, urban-space constraints, and population-level mobility distributions. In that usage, CASMOPOLITAN functions as a conceptual comparator rather than as the same algorithmic object. CAMS is a trajectory-simulation framework built around MobExtractor, GeoGenerator, and TrajEnhancer, whereas CASMOPOLITAN, in the evidence available here, is a Bayesian optimization method for categorical and mixed spaces (Du et al., 16 Jun 2025). The comparison therefore points to an analogy in staged, locality-aware, structure-sensitive reasoning, not to identity of method.

Taken together, the available record portrays CASMOPOLITAN as a high-dimensional mixed-space BO optimizer whose core contribution lies in its two-part trust-region design, transformed overlapped GP kernel, ordinal encoding for GP modeling, and Thompson-sampling-based acquisition workflow (Ngo et al., 9 Aug 2025). The most specific contemporary insight is that these components can be preserved while the local search region is redefined through an encoded-space Mahalanobis ellipsoid plus a Hamming-distance constraint, yielding a method that outperforms vanilla CASMOPOLITAN on most tested benchmarks (Ngo et al., 9 Aug 2025).