A Decoupled Basis-Vector-Driven Generative Framework for Dynamic Multi-Objective Optimization

Yaoming Yang; Shuai Wang; Bingdong Li; Peng Yang; Ke Tang

cs.LGApr 1, 2026

A Decoupled Basis-Vector-Driven Generative Framework for Dynamic Multi-Objective Optimization

Authors: Yaoming Yang, Shuai Wang, Bingdong Li, Peng Yang, Ke Tang

Abstract

Dynamic multi-objective optimization requires continuous tracking of moving Pareto fronts. Existing methods struggle with irregular mutations and data sparsity, primarily facing three challenges: the non-linear coupling of dynamic modes, negative transfer from outdated historical data, and the cold-start problem during environmental switches. To address these issues, this paper proposes a decoupled basis-vector-driven generative framework (DB-GEN). First, to resolve non-linear coupling, the framework employs the discrete wavelet transform to separate evolutionary trajectories into low-frequency trends and high-frequency details. Second, to mitigate negative transfer, it learns transferable basis vectors via sparse dictionary learning rather than directly memorizing historical instances. Recomposing these bases under a topology-aware contrastive constraint constructs a structured latent manifold. Finally, to overcome the cold-start problem, a surrogate-assisted search paradigm samples initial populations from this manifold. Pre-trained on 120 million solutions, DB-GEN performs direct online inference without retraining or fine-tuning. This zero-shot generation process executes in milliseconds, requiring approximately 0.2 seconds per environmental change. Experimental results demonstrate that DB-GEN improves tracking accuracy across various dynamic benchmarks compared to existing algorithms.

Figures & tables

Fig. 1: The overall framework of the proposed DB-GEN.

Problem	$n_{t}$	$\tau_{t}$	STT-MOEA/D	DIP-DMOEA	VARE	SIKT-DMOEA	Ours
DF1	5	10	7.36e-02(8.78e-03) $-$	3.71e-02(3.15e-04) $-$	4.22e-02(3.33e-03) $-$	5.64e-02(8.69e-03) $-$	1.31e-02(4.08e-03)
	10	5	6.75e-01(3.59e-03) $-$	6.48e-02(5.87e-03) $-$	1.24e-01(1.69e-02) $-$	1.50e-01(2.38e-02) $-$	2.10e-02(9.62e-04)
	10	10	6.33e-01(7.77e-02) $-$	2.33e-02(4.03e-03) $-$	2.85e-02(1.85e-03) $-$	5.35e-02(7.42e-03) $-$	1.05e-02(2.22e-04)
DF2	5	10	2.19e-02(1.38e-02) $\approx$	4.97e-02(6.99e-03) $-$	6.51e-02(5.91e-03) $-$	4.21e-02(4.36e-03) $-$	2.15e-02(5.04e-03)
	10	5	4.02e-01(6.23e-03) $-$	9.55e-02(8.71e-03) $-$	1.36e-01(9.32e-02) $-$	1.08e-01(7.67e-03) $-$	4.69e-02(1.64e-03)
	10	10	3.85e-01(3.81e-03) $-$	3.82e-02(8.73e-04) $-$	4.22e-02(3.67e-03) $-$	3.63e-02(3.91e-03) $-$	1.57e-02(1.06e-03)

TABLE I: The MIGD values and standard deviation in parentheses obtained by SOTAs and Ours.

Problem	$n_{t}$	$\tau_{t}$	STT-MOEA/D	DIP-DMOEA	VARE	SIKT-DMOEA	Ours
FDA1	5	10	1.07e-01(4.61e-03) $-$	5.31e-01(1.68e-03) $-$	6.01e-01(1.57e-03) $-$	6.17e-01(1.49e-02) $-$	6.27e-01(6.56e-03)
	10	5	7.13e-02(3.59e-03) $-$	6.51e-01(4.69e-02) $-$	6.56e-01(6.34e-03) $-$	5.08e-01(4.21e-02) $-$	6.74e-01(3.47e-03)
	10	10	1.02e-01(3.55e-03) $-$	7.19e-01(4.46e-03) $+$	8.07e-01(5.19e-04) $+$	6.60e-01(3.99e-03) $\approx$	6.75e-01(4.19e-03)
FDA2	5	10	1.74e-03(3.06e-03) $-$	5.54e-01(3.50e-03) $-$	6.84e-01(1.08e-02) $-$	5.64e-01(1.69e-03) $-$	2.00e+00(1.14e-02)
	10	5	1.33e-02(1.79e-03) $-$	7.11e-01(1.46e-02) $-$	6.07e-01(2.29e-04) $-$	5.57e-01(1.35e-02) $-$	1.82e+00(2.01e-02)
	10	10	2.63e-01(1.68e-03) $-$	7.90e-01(4.57e-03) $-$	5.85e-01(5.56e-04) $-$	5.65e-01(1.71e-03) $-$	1.96e+00(2.85e-02)

TABLE II: The MHV values and standard deviation in parentheses obtained by SOTAs and Ours.

Fig. 2: The IGD variation curves over environments.

Panel of Figure 5 — Fig. 3: Zero-shot solution generation upon specific environmental change on the DF1 problem. (a) Proposed latent centroid perturbation. (b) Perturbation on all previous solutions.

Fig. 4: Effect of historical data scale on the mean IGD and the average rank.

Fig. 5: Correlation between the normalized latent space distance of new environments and the average IGD improvement rate.

Variant	MIGD	Degradation ( $\Delta$ )	$+/-/\approx$
Full Model	0.0832	-	-
w/o High-Low Freq	0.0983	+18.1%	4 / 53 / 0
w/o Basis Learning	0.1205	+44.8%	14 / 43 / 0
w/o Solution VAE	0.1028	+23.6%	23 / 34 / 0
w/o Triplet Loss	0.1048	+26.0%	15 / 42 / 0
w/o Classify Loss	0.0934	+12.3%	17 / 40 / 0

TABLE III: Ablation study results on the core mechanisms.

Fig. 6: The t-SNE visualization of the environment-level structural latent representations (

\mathbf{z}_{struct}

) generated by the pre-trained model on unseen test problems.

Fig. 7: The t-SNE visualization of DF6 and Problem 64/67.

Fig. 8: Visualization of the generated solutions under different masking strategies across environments on the DF6 problem.

Parameter	Tested Values and Mean IGD
Basis Vectors ( $K$ )	32	64	128	256	384	512
Basis Vectors ( $K$ )	0.1048	0.0975	0.1046	0.0832	0.0854	0.0872
Perturbation Radius ( $\sigma$ )	0.05	0.10	0.15	0.20	0.25	0.30
Perturbation Radius ( $\sigma$ )	0.1040	0.0923	0.0970	0.0832	0.0886	0.0871
Candidate Pool ( $N_{cand}$ )	50,000	100,000	150,000	200,000	250,000	300,000
Candidate Pool ( $N_{cand}$ )	0.1140	0.0910	0.0920	0.0832	0.0870	0.0840

TABLE IV: Parameter sensitivity analysis result.

Problem	Ours	STT-MOEA/D	DIP-DMOEA	SIKT-DMOEA	VARE
DRA	1.35E-02	9.18E-02	3.63E-02	1.22E-01	5.74E-02
DPP	7.19E-02 $\approx$	1.58E-01	2.90E-01	1.77E-01	7.07E-02

TABLE V: MIGD results on real-world dynamic problems.

Fig. 9: Comparison of execution time (in seconds) across the DF test suite when

\tau_{t}=10

and

n_{t}=10

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Category	Parameter	Value
DMOP Benchmarks	Number of Objectives ( $M$ )	2, 3
	Decision Variables Dimension ( $D$ )	10
	Change Frequency & Severity ( $\tau_{t},n_{t}$ )	(10,10), (5,10), (10,5)
	Number of Environments	30
	Total Generations per Run	$30\times\tau_{t}$
DB-GEN Architecture	Latent Space Dimension ( $d_{z}$ )	64

Appendix

TABLE I : Parameter Configurations and Implementation Environments

Panel of Figure 16 — Fig. 1 : Complete tracking performance of the proposed DB-GEN utilizing the centroid perturbation strategy across all 30 environments on the DF1 problem. The grey dots delineate the true PF, the blue dots represent the un-evaluated inherited population, the red star marks the latent centroid, and the orange dots denote the generated initial candidates.

Panel of Figure 17 — Fig. 2 : Complete tracking performance utilizing the alternative all-point perturbation strategy across all 30 environments on the DF1 problem.

Panel of Figure 18 — Fig. 3 : Complete IGD variation curves of different algorithms over consecutive environments on all DF test instances.

Fig. 4 : The dynamic Pareto Front (PF) of the DF6 benchmark, illustrating the non-linear curvature changes and localized disconnected regions across different environmental states.

Fig. 5 : Latent space visualization of DF6, Problem 64, and Problem 67. Although distributed in distant sub-regions, the trajectory of DF6 is spatially positioned between Problem 64 and Problem 67, reflecting its compositional dynamic properties.

Problem	$n_{t}$	$\tau_{t}$	STT-MOEA/D	DIP-DMOEA	VARE	SIKT-DMOEA	Ours
DF1	5	10	9.30e-02(3.84e-03) $-$	4.72e-01(3.09e-03) $-$	5.81e-01(1.56e-03) $-$	4.55e-01(1.21e-02) $-$	6.09e-01(2.74e-03)
	10	5	7.91e-02(8.64e-03) $-$	4.34e-01(4.66e-04) $-$	5.39e-01(4.63e-03) $-$	3.73e-01(1.82e-02) $-$	5.58e-01(1.74e-03)
	10	10	9.15e-02(8.24e-04) $-$	4.95e-01(4.53e-02) $-$	4.43e-01(2.90e-03) $-$	4.46e-01(7.95e-03) $-$	5.15e-01(4.57e-04)
DF2	5	10	2.63e-01(2.99e-03) $-$	6.32e-01(7.12e-03) $+$	6.55e-01(3.61e-03) $+$	6.56e-01(7.68e-03) $+$	6.24e-01(4.49e-03)
	10	5	2.45e-01(7.95e-04) $-$	5.81e-01(5.35e-04) $-$	5.51e-01(1.20e-03) $-$	5.84e-01(7.65e-03) $-$	6.28e-01(2.90e-03)
	10	10	2.47e-01(6.28e-04) $-$	6.71e-01(4.58e-04) $-$	6.76e-01(6.94e-03) $-$	6.67e-01(5.75e-03) $-$	6.81e-01(1.44e-03)

Appendix

TABLE II : The MHV values and standard deviation in parentheses obtained by SOTAs and Ours.

Fig. 6 : The $p$ -value heatmap of the pairwise Wilcoxon signed-rank tests across different historical data scales. Cells annotated with $*$ indicate a statistically significant difference ( $p<0.05$ ). The scale units are $\times 10^{7}$ .

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Training-Free Diffusion-Driven Modeling of Pareto Set Evolution for Dynamic Multiobjective Optimization

Jian Guan, Huolong Wu, Zhenzhong Wang, Gary G. Yen+1

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Multi-Objective Optimization Multi-Objective Evolutionary Optimization

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Dynamic multiobjective optimization problems (DMOPs) feature time-varying objectives, which cause the Pareto optimal solution (POS) set to drift over time and make it difficult to maintain both convergence and diversity under limited response time. Many existing prediction-based dynamic multiobjective evolutionary algorithms (DMOEAs) either depend on learned models with nontrivial training cost or employ one-step population mapping, which may overlook the gradual nature of POS evolution. This paper proposes DD-DMOEA, a training-free diffusion-based dynamic response mechanism for DMOPs. The key idea is to treat the POS obtained in the previous environment as a "noisy" sample set and to guide its evolution toward the current POS through an analytically constructed multi-step denoising process. A knee-point-based auxiliary strategy is used to specify the target region in the new environment, and an explicit probability-density formulation is derived to compute the denoising update without neural training. To reduce the risk of misleading guidance caused by knee-point prediction errors, an uncertainty-aware scheme adaptively adjusts the guidance strength according to the historical prediction deviation. Experiments on the CEC2018 dynamic multiobjective benchmarks show that DD-DMOEA achieves competitive or better convergence-diversity performance and provides faster dynamic response than several state-of-the-art DMOEAs.

Abstract

Multi-Party Multi-Objective Optimization

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