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This paper introduces Parametric Pareto Set Learning with Multi-Objective Bayesian Optimization (PPSL-MOBO), a novel framework designed to solve expensive multi-objective optimization problems where optimal solutions must adapt to varying parameters. The core idea is to learn a unified neural architecture that maps both preferences and parameters to Pareto-optimal solutions, leveraging a hypernetwork with Low-Rank Adaptation (LoRA) for efficient parametric variation capture, and integrating Gaussian process surrogates with hypervolume-based acquisition to minimize expensive function evaluations. PPSL-MOBO demonstrates effectiveness on multi-objective optimization with shared components and dynamic multi-objective optimization, achieving competitive performance with significantly fewer function evaluations and faster inference compared to baseline methods.

The paper addresses several challenges in Parametric Multi-Objective Optimization (PMO) for expensive objective evaluations:

Computational Intractability: Traditional methods require re-solving the entire optimization problem for each new parameter value, making real-time adaptation impossible.

Proposed a unified model architecture that learns parametric PSs as a continuous function of both preferences and exogenous parameters, enabling real-time multi-objective decision making without retraining.
Developed a sample-efficient learning framework that integrates flexible surrogate models with intelligent acquisition strategies, dramatically reducing the number of expensive evaluations required.
Applied PPSL-MOBO to modular design scenarios, where certain decision variables are shared across families of solutions, enabling efficient, real-time design for modular and customizable systems.
Extended PPSL-MOBO to dynamic multi-objective optimization, demonstrating efficient adaptation to temporal changes by treating time as a parametric context.

The proposed PPSL-MOBO framework operates through three tightly coupled components: a hypernetwork-LoRA architecture, a Gaussian process-based training scheme, and an intelligent data acquisition strategy.

Model Architecture The method extends existing PSL models, which map a preference vector λ to a solution x on the PS (x = h_θps(λ)), to handle parametric objectives where the PS is a function of an external context or task parameter t. Instead of a naive hypernetwork directly generating all PS model weights θps, which can be high-dimensional, PPSL-MOBO uses a Low-Rank Adaptation (LoRA) approach. For each layer l of the PS model, its weights θps^l(t) are decomposed into a shared base component θ0^l and a low-rank matrix product B^l(t)A^l(t): θps^l(t) = θ0^l + B^l(t)A^l(t) Here, θ0^l is a trainable base parameter set shared across all tasks. The matrices B^l(t) ∈ R^(de×r) and A^l(t) ∈ R^(r×ke) are low-rank adapters, where r is a small integer (r < min(de, ke)). The hypernetwork then generates only the compact low-rank matrices (Olora(t) = gohn(t)), significantly reducing its output dimensionality.

Training Framework with Gaussian Process The goal is to train the shared base weights θ0 and the hypernetwork weights θhn such that the model's output x = h_θps(t)(λ) is a Pareto-optimal solution for any given task parameter t and preference vector λ. This is formulated as minimizing the expected Smooth Tchebycheff (STCH) loss over distributions of tasks (Pt) and preferences (Pλ): L(θhn, θ0) = E_t~~Pt,λ~~Pλ [lstch (h_θps(t)(λ) | λ, t)] Since true objective functions f(x;t) are expensive, surrogate models are used. An augmented input space Z = X × T is defined, where z = [x, t]. Independent Gaussian Process (GP) models are constructed for each objective function fi over this augmented space: fi(z) ~ GP(μi(z), ki(z, z')) The GP hyperparameters are optimized by maximizing the marginal log-likelihood. To balance exploitation and exploration, the Lower Confidence Bound (LCB) is used as the acquisition signal, defining a surrogate objective vector f for a given pair (x, t): f(x;t) = μ(z) – βσ(z) where β is a hyperparameter. This leads to the final, differentiable surrogate loss function: L(θhn, θ0) = E_t~~Pt,λ~~Pλ [Îstch (h_θps(t)(λ) | λ,t)] where Îstch is the surrogate STCH loss for a given task t and preference λ, defined as: Îstch(x | λ, t) = (1/ν) log(Σ_j=1^m e^(λ_j(f_j(x;t) - (z_j - ε)))) The expectation is approximated using Monte Carlo sampling, and gradients are backpropagated to update θ0 and θhn.

Intelligent Data Acquisition At each iteration, a small batch of new task-solution pairs (x, t) is selected.

Candidate Pool Generation: The trained parametric PS model h_θps(t)(λ) generates a large pool of P high-quality candidate points by sampling task-preference pairs {(tp, λp)}p=1^P.
Batched Selection via Hypervolume Improvement (HVI): From the candidate pool, a batch of B points is selected for expensive evaluation using a sequential greedy strategy based on HVI (Eq. (6)). The HVI is calculated using the LCB value from the augmented-space GP surrogates.

Parametric Multi-Objective Optimization Problem: F*(t) = arg min_x (f1(x, t), f2(x, t), ..., fm(x,t)) where x is the decision variable vector in decision space X ⊆ R^n, t denotes parameters in space Θ ⊆ R^p, and F*(t) defines optimal solutions.
Pareto Set Learning Model: x = h_θ(λ), where λ ∈ Δ^(m-1) is a preference vector and h_θ: Δ^(m-1) → X is a learnable model parameterized by θ.
Aggregation Function: Smooth Tchebycheff (STCH) aggregation is adopted as the primary method: lstch(x | λ) = (1/ν) log(Σ_j=1^m e^(λ_j(f_j(x) - (z_j - ε)))) where ν > 0 is a smoothing parameter, z_j < min_x∈X f_j(x) is the ideal value for the j-th objective, and ε > 0 is a small constant.
Hypernetwork-LoRA Architecture: For each layer l of the PS model, weights are decomposed as θps^l(t) = θ0^l + B^l(t)A^l(t), where B^l(t) ∈ R^(de×r) and A^l(t) ∈ R^(r×ke) are low-rank adapters, and r is a small integer (r < min(de, ke)). Olora(t) represents the collection of all entries in A^l(t) and B^l(t) across all layers.
Gaussian Process (GP) Models: Independent GP models are constructed for each objective function fi over an augmented input space Z = X × T, where z = [x, t]. The GP is defined as fi(z) ~ GP(μi(z), ki(z, z')).
Acquisition Function: Lower Confidence Bound (LCB) is used as the acquisition signal: f(x;t) = μ(z) – βσ(z), where β is a hyperparameter controlling the exploration-exploitation trade-off.
Data Acquisition: A batch of B points is selected using a sequential greedy strategy based on Hypervolume Improvement (HVI), calculated using the LCB values from GP surrogates.
Optimization: The GP hyperparameters are optimized by maximizing the marginal log-likelihood on the observed data. The global training objective is optimized via gradient descent, with expectations approximated using Monte Carlo sampling.

Datasets:
- Multi-objective optimization with shared components: Real-world multi-objective test problems suite (Tanabe and Ishibuchi 2020; Lin et al. 2022), specifically problems RE21, RE33, RE37.
- Dynamic multi-objective optimization: Benchmark problems DF1 and DF2 (Jiang et al. 2018).
Metrics:
- Hypervolume (HV) values.
- Runtime (in seconds).
Baselines:
- qParEGO
- qEHVI
- PSL-MOBO (a variant without the parametric component, likely for comparison of the core PSL idea)
- NSGA-II (for shared components and dynamic MOO)
- DNSGA-II (for dynamic MOO)
Numeric Results:
- HV values on MOPs with shared components (Table 1):
  - RE21 (x1): PPSL-MOBO 7.33e-01 (2.31e-03), qEHVI 7.34e-01 (5.79e-05), qParEGO 6.96e-01 (2.89e-02), NSGA-II 6.52e-01 (3.57e-03).
  - RE21 (x2): PPSL-MOBO 7.79e-01 (7.91e-03), qEHVI 7.84e-01 (3.43e-05), qParEGO 7.53e-01 (5.45e-03), NSGA-II 6.90e-01 (1.04e-02).
  - RE21 (x3): PPSL-MOBO 6.68e-01 (7.07e-03), qEHVI 6.76e-01 (8.41e-05), qParEGO 6.53e-01 (6.11e-03), NSGA-II 6.14e-01 (2.74e-03).
  - RE21 (x4): PPSL-MOBO 7.99e-01 (2.74e-03), qEHVI 8.02e-01 (4.01e-05), qParEGO 7.70e-01 (1.15e-02), NSGA-II 7.02e-01 (2.24e-03).
  - RE21 (x1, x2): PPSL-MOBO 6.23e-01 (7.84e-04), qEHVI 6.23e-01 (1.64e-05), qParEGO 6.19e-01 (1.77e-03), NSGA-II 5.92e-01 (5.35e-03).
  - RE21 (x2, x3): PPSL-MOBO 5.69e-01 (6.26e-05), qEHVI 5.67e-01 (7.38e-05), qParEGO 5.58e-01 (4.16e-03), NSGA-II 5.38e-01 (1.38e-03).
  - RE21 (x3, x4): PPSL-MOBO 5.66e-01 (7.15e-04), qEHVI 5.65e-01 (1.86e-04), qParEGO 5.60e-01 (9.90e-04), NSGA-II 5.36e-01 (1.81e-03).
  - RE21 (x1,x2,x3): PPSL-MOBO 4.00e-01 (6.61e-06), qEHVI 3.99e-01 (7.76e-05), qParEGO 3.98e-01 (1.21e-03), NSGA-II 3.96e-01 (1.52e-04).
  - RE21 (x2, x3, x4): PPSL-MOBO 5.19e-01 (2.19e-05), qEHVI 5.18e-01 (7.88e-05), qParEGO 5.15e-01 (3.12e-03), NSGA-II 5.11e-01 (1.43e-03).
- Runtime comparison (in seconds) (Table 2):
  - RE21: PPSL-MOBO Training 364.4s, PPSL-MOBO Inference 0.019s. Baselines: qParEGO 317.1s, qEHVI 551.3s, PSL-MOBO 1761.3s.
  - RE33: PPSL-MOBO Training 581.8s, PPSL-MOBO Inference 0.040s. Baselines: qParEGO 430.9s, qEHVI 1038.9s, PSL-MOBO 1812.4s.
  - RE37: PPSL-MOBO Training 639.0s, PPSL-MOBO Inference 0.028s. Baselines: qParEGO 349.2s, qEHVI 873.1s, PSL-MOBO 2668.9s.
Runtime/Training Cost: PPSL-MOBO was trained only once using a total of 200 evaluations for shared component problems, whereas baseline methods required re-execution for each parameter configuration with a budget of 100 evaluations per task, totaling 1,000 evaluations. PPSL-MOBO provides near-instantaneous inference (in milliseconds) for new parameter values.

No explicit paper-reported limitations were found.

Method evidence: "This work proposes Parametric Pareto Set Learning with Multi-Objective Bayesian Optimization (PPSL-MOBO), a novel framework that learns a unified mapping from both preferences and parameters to Pareto-optimal solutions. PPSL-MOBO leverages a hypernetwork with Low-Rank Adaptation (LoRA) to efficiently capture parametric variations, while integrating Gaussian process surrogates and hypervolume-based acquisition to minimize expensive function evaluations." (section: Abstract)
Result evidence: "The experimental results in Tables 1 demonstrate the effectiveness and efficiency of our proposed method. PPSL-MOBO achieves competitive hypervolume performance across different shared component configurations, often matching or outperforming baseline methods that are specifically optimized for individual tasks. The most significant advantage of PPSL-MOBO lies in its computational and sample efficiency. While baseline methods must be re-executed from scratch for each new shared component configuration, PPSL-MOBO requires only a one-time training phase followed by near-instantaneous inference (in milliseconds) for new parameter values." (section: 4.2 Results Analysis)
Limitation evidence: Not stated in the paper.

Parametric Pareto Set Learning for Expensive Multi-Objective Optimization

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