TSEMO: Efficient Multi-Objective Optimization

• The paper introduces TSEMO, which combines independent Gaussian processes, Thompson sampling, and evolutionary search to iteratively sample Pareto fronts.
• TSEMO is a multi-objective Bayesian optimization method targeting continuous design spaces, with potential but costly application in discrete settings.
• TSEMO’s sample-then-search approach balances exploration and exploitation by drawing posterior sample functions and using evolutionary algorithms to find Pareto-optimal solutions.

Thompson Sampling Efficient Multi-objective Optimization (TSEMO) is a multi-objective Bayesian optimization method that combines Gaussian-process surrogates, Thompson sampling, and multi-objective evolutionary search for expensive black-box optimization. In the characterization used by recent work on alloy design, TSEMO models each objective with an independent Gaussian process, draws posterior sample functions, searches the sampled objective surface for a Pareto set by means of a genetic or evolutionary algorithm, and selects new evaluations from that sampled front; iterating this procedure progressively approximates the true Pareto front (Mamun et al., 2024). Later work places TSEMO within a broader family of Thompson-sampling-based multi-objective optimizers and contrasts it with hypervolume-driven and Pareto-optimal Thompson-sampling variants (Renganathan et al., 2023).

1. Definition and core mechanism

TSEMO is a GP-based multi-objective Bayesian optimization procedure. In the account summarized in the alloy-design study, its surrogate model consists of independent Gaussian processes for each objective $f_k(x)$. Thompson sampling is applied by drawing one posterior sample function per objective,

$$\tilde{\mathbf{f}}(\cdot)=\big(\tilde f_1(\cdot),\dots,\tilde f_M(\cdot)\big),$$

so that the optimizer works with a sampled realization of the full vector-valued objective rather than with posterior means alone (Mamun et al., 2024).

The search stage is explicitly multi-objective. For each sampled realization, an evolutionary algorithm, such as NSGA-II or a genetic algorithm, is used to search the design space for a Pareto set under the sampled objectives. Newly evaluated points are then selected from the Pareto set, or from a subset of it, returned by that evolutionary search. Repeating this sequence produces a succession of sampled Pareto fronts and uses them to guide data acquisition (Mamun et al., 2024).

Within that formulation, TSEMO is designed for continuous search spaces. The alloy-design paper further states that it can in principle be applied to discrete spaces, but at high computational cost, a point that becomes important in catalogue-based materials design and other finite-candidate settings (Mamun et al., 2024).

2. Thompson-sampling interpretation

The distinctive feature of TSEMO is that it performs Thompson sampling at the level of the entire Pareto structure rather than at the level of a scalar utility. The evolutionary optimization over $\tilde{\mathbf f}$ is equivalent to computing a Pareto front of the sampled objective surface, and that sampled front functions as a Thompson sample of the unknown true Pareto front (Mamun et al., 2024).

This makes TSEMO a “sample-then-search” method. Exploration arises because posterior samples can place Pareto-optimal trade-offs in regions that are still weakly constrained by data, while exploitation arises because as the posterior contracts, sampled fronts concentrate near the true front. A plausible implication is that TSEMO’s behavior is governed less by a closed-form acquisition landscape than by the quality of the internal evolutionary solution of each sampled multi-objective problem.

The same structural idea appears in later Pareto-oriented Thompson-sampling work. qPOTS, for example, also fits independent GPs for each objective, samples all objectives jointly, and solves a sampled multi-objective problem by NSGA-II; its contribution is to interpret the policy explicitly as sampling according to the probability that points are Pareto optimal, rather than routing selection through hypervolume calculations (Renganathan et al., 2023). This comparison clarifies the sense in which TSEMO is a canonical Thompson-sampling approach to multi-objective BO: it relies on posterior sample paths and Pareto search, not on analytic hypervolume improvement formulas.

3. Computational profile and limitations

In the alloy-design study, TSEMO is grouped with computationally heavy multi-objective BO methods. The reported sources of cost are the need to train a GP for each objective, draw Thompson samples, and then solve a nontrivial evolutionary search problem over the sampled objective surface at every BO iteration (Mamun et al., 2024).

The same study characterizes original TSEMO as essentially sequential and not natively designed for $q$-batch acquisition optimization. Although multiple points can be extracted heuristically from a sampled Pareto set, the method is not presented there as optimizing a joint batch acquisition with explicit diversity guarantees. This contrasts with later $q$-methods that optimize batch selection directly (Mamun et al., 2024).

Scaling to large discrete catalogues is a particular weakness in that characterization. For candidate sets containing $13{,}700$ CALPHAD-based alloy instances or roughly $3{,}700$ experimental instances, a GA-based inner search is possible but may be inefficient relative to approaches that simply evaluate an acquisition score on all candidates. The paper therefore treats TSEMO as applicable in principle, but not well matched to a discrete design space in which exhaustive acquisition ranking is feasible (Mamun et al., 2024).

Noise handling is another limitation in that comparison. The alloy paper emphasizes that $q$NEHVI integrates over uncertainty in the latent Pareto set implied by noisy observations, whereas TSEMO, as originally formulated there, samples objectives directly but does not integrate over uncertainty in the current Pareto set in the same way. This suggests reduced robustness when the observational noise is substantial (Mamun et al., 2024).

4. Position relative to hypervolume-based multi-objective BO

Recent hypervolume-based methods are the main reference point against which TSEMO is discussed in contemporary materials optimization. The alloy-design study states that expected hypervolume-based geometrical acquisition functions have demonstrated superior performance and speed compared to algorithms such as TSEMO and parEGO, and uses that observation to motivate the evaluation of $q$EHVI, $q$NEHVI, $$\tilde{\mathbf{f}}(\cdot)=\big(\tilde f_1(\cdot),\dots,\tilde f_M(\cdot)\big),$$0parEGO, and $$\tilde{\mathbf{f}}(\cdot)=\big(\tilde f_1(\cdot),\dots,\tilde f_M(\cdot)\big),$$1NparEGO rather than TSEMO itself (Mamun et al., 2024).

The distinction is methodological. TSEMO samples objective functions and searches their sampled Pareto front. By contrast, $$\tilde{\mathbf{f}}(\cdot)=\big(\tilde f_1(\cdot),\dots,\tilde f_M(\cdot)\big),$$2EHVI and $$\tilde{\mathbf{f}}(\cdot)=\big(\tilde f_1(\cdot),\dots,\tilde f_M(\cdot)\big),$$3NEHVI directly target expected hypervolume improvement, use Monte Carlo integration over the joint GP posterior, support autodiff-based optimization, and are explicitly parallel. In the same paper, original EHVI, TSEMO, and parEGO are all described as costly and lacking parallel support, while $$\tilde{\mathbf{f}}(\cdot)=\big(\tilde f_1(\cdot),\dots,\tilde f_M(\cdot)\big),$$4EHVI and $$\tilde{\mathbf{f}}(\cdot)=\big(\tilde f_1(\cdot),\dots,\tilde f_M(\cdot)\big),$$5NEHVI are introduced precisely to address those issues (Mamun et al., 2024).

For discrete alloy design, the paper’s practical recommendation is clear: use $$\tilde{\mathbf{f}}(\cdot)=\big(\tilde f_1(\cdot),\dots,\tilde f_M(\cdot)\big),$$6EHVI for noise-free computational problems and $$\tilde{\mathbf{f}}(\cdot)=\big(\tilde f_1(\cdot),\dots,\tilde f_M(\cdot)\big),$$7NEHVI for noisy experimental problems. TSEMO is not implemented or benchmarked there; it appears as part of the historical and methodological landscape and is implicitly treated as a less attractive option for that particular application domain (Mamun et al., 2024).

A closely related comparison appears in qPOTS. That work describes Bradford/TSEMO-like algorithms as combining GPs with sampled objectives and NSGA-II-like inner optimization, but emphasizes that hypervolume calculations on sampled fronts can be expensive and that batch selection becomes difficult as $$\tilde{\mathbf{f}}(\cdot)=\big(\tilde f_1(\cdot),\dots,\tilde f_M(\cdot)\big),$$8 grows. qPOTS therefore removes hypervolume from the acquisition stage and selects a diverse batch from the sampled Pareto set by a maximin distance rule (Renganathan et al., 2023).

5. Related Thompson-sampling multi-objective methods

TSEMO sits within a broader Thompson-sampling landscape rather than standing alone. Among the nearest relatives, qPOTS is the most direct analogue: it retains independent GPs, posterior function sampling, and multi-objective evolutionary optimization on sampled objectives, but frames the procedure as Pareto Optimal Thompson Sampling and avoids hypervolume in the acquisition step (Renganathan et al., 2023).

A different extension appears in preferential multi-objective Bayesian optimization. Dueling Scalarized Thompson Sampling (DSTS) combines independent GP surrogates, random augmented Chebyshev scalarizations, and Thompson sampling under objective-wise Logistic preference feedback. It can be viewed as a preferential analogue of TS-plus-scalarization multi-objective BO and proves asymptotic consistency in the finite-$$\tilde{\mathbf{f}}(\cdot)=\big(\tilde f_1(\cdot),\dots,\tilde f_M(\cdot)\big),$$9 setting for a modified policy, thereby showing that Thompson-sampling-based Pareto exploration remains meaningful even when direct objective values are unavailable (Astudillo et al., 2024).

Bandit-theoretic analysis also clarifies why TSEMO-style modeling of vector objectives is important. In multi-objective bandits with a preference function, Durand and Gagné show that one cannot simply scalarize noisy vector outcomes into a single-objective problem: $\tilde{\mathbf f}$0 can differ from $\tilde{\mathbf f}$1. Their analysis supports the design principle of modeling objectives directly and applying preferences or scalarizations to posterior means or posterior samples, rather than modeling only scalarized observations (Durand et al., 2017).

Other uses of Thompson sampling in multi-objective optimization are structurally distinct from TSEMO. Dynamic Thompson sampling for MOEA/D treats reproduction operators as bandit arms and updates Beta-Bernoulli posteriors to select operators online under non-stationary rewards; here Thompson sampling adapts the internal dynamics of an evolutionary algorithm rather than defining the outer Bayesian optimizer itself (Sun et al., 2020). In contextual bandits, MOL-TS samples objective-specific linear parameters and selects from an effective Pareto front that accounts for repeated selections over time, achieving a worst-case Pareto regret bound of $\tilde{\mathbf f}$2; this is a different setting, but it illustrates a history-aware Pareto archive concept absent from standard TSEMO formulations (Park et al., 30 Nov 2025).

6. Applications, status in materials design, and prospective directions

In current materials-design literature, TSEMO is primarily a reference method rather than the preferred operational choice. The alloy-design paper explicitly states that TSEMO is only mentioned as part of the broader multi-objective BO landscape and is neither implemented nor benchmarked. Its role there is diagnostic: it marks an earlier GP-based, Thompson-sampling, evolutionary-search tradition against which hypervolume-based batch methods are judged more suitable for discrete aluminium-alloy optimization (Mamun et al., 2024).

This does not make TSEMO obsolete in every setting. The same material suggests that its combination of GP surrogates, Thompson sampling, and evolutionary Pareto search remains conceptually appealing, particularly when the design space is continuous, evolutionary search is a natural inner solver, and the objective is to obtain diverse Pareto samples rather than to optimize hypervolume directly. A plausible implication is that TSEMO remains relevant where sampled-front diversity matters more than exact alignment with a hypervolume metric.

Recent single-objective GP-TS work also points to a possible improvement path. Rootfinding-based global optimization of GP posterior samples exploits a decoupled representation of the posterior sample and uses structured starting points for multi-start gradient methods. That work is not a TSEMO paper, but it explicitly discusses how such inner-loop optimization techniques could inform TSEMO by improving the optimization of sampled surrogate functions and by supplying better seeds for multi-objective search (Adebiyi et al., 2024). This suggests that some of TSEMO’s practical limitations may stem as much from the optimization of sampled functions as from Thompson sampling itself.

A further prospective direction is front-centric uncertainty quantification. Top-Two Pareto Front Thompson Sampling for multi-objective bandits introduces an anytime Bayesian PSI algorithm together with a Bhattacharyya-based uncertainty metric that measures overlap between first and second Pareto fronts. A plausible implication is that analogous front-to-front uncertainty measures could complement TSEMO-like continuous MOBO by providing stopping or monitoring criteria for the stability of the current Pareto approximation (Saerens et al., 17 Jun 2026).

Taken together, these developments place TSEMO in a precise methodological position. It is a foundational Thompson-sampling multi-objective Bayesian optimizer built around GP surrogates and sampled Pareto search. Contemporary work does not reject that architecture; rather, it refines it in three directions: direct hypervolume optimization for batch and noisy settings, explicit Pareto-optimal Thompson-sampling rules that avoid hypervolume calculations, and more specialized posterior-sampling schemes for preferences, contextual decision-making, or front identification (Mamun et al., 2024, Renganathan et al., 2023).