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

Jian Guan; Huolong Wu; Zhenzhong Wang; Gary G. Yen; Min Jiang

cs.NEMar 23, 2026

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

Authors: Jian Guan, Huolong Wu, Zhenzhong Wang, Gary G. Yen, Min Jiang

Organizations: Institute of Artificial Intelligence, Xiamen University, Xiamen, 361005, China · Department of Artificial Intelligence, School of Informatics, Xiamen University; the Key Laboratory of Multimedia Trusted Perception and Efficient Computing, Ministry of Education; and the Key Laboratory of Digital Protection and Intelligent Processing of Intangible Cultural Heritage of Fujian and Taiwan, Ministry of Culture and Tourism, Xiamen, 361005, China · Department of Artificial Intelligence, Sichuan University, Chengdu, 610065, China

Abstract

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.

Figures & tables

Fig. 1: Two evolutionary modes of the POS in dynamic environments . (a) The single-step, direct mapping employed by existing prediction methods. (b) The gradual and smooth evolutionary process of the POF in the real world. (c) Correspondingly, in the decision space, the evolution of the POS exhibits a continuous manifold pattern. The red dashed box ( $\Delta t$ ) indicates the key intermediate transitional states often ignored by the methods in (a).

Fig. 2: Overview of the proposed DD-DMOEA. (a) The AKP strategy partitions the objective space and predicts the positions of the knee points ( $k^{t}_{i}$ ) at the moment $t$ . This prediction assists in generating an approximate target POS distribution based on $POS_{t-1}$ . (b) The DDM treats the $POS_{t-1}$ as a noisy distribution, and a training-free diffusion model executes an efficient and progressive denoising process. (c) As the environment changes ( $t$ -1 $\to$ $t$ ), this denoising process gradually transforms the “noisy” old population into a refined one adapted to the new environment, enabling rapid tracking of the POF’s evolution ( $POF_{t-1}\to POF_{t}$ ).

Fig. 3: Illustration of dividing objection space by the knee point. The number of subspaces here is 3.

Prob.	( $n_{t},\tau_{t}$ )	KT-DMOEA	IGP-DMOEA	DIP-DMOEA	KTM-DMOEA	PSGN	VARE	DM-DMOEA	DD-DMOEA
DF1	(10, 10)	0.0402±9.53e-03(+)	0.0248±3.59e-03(+)	0.0264±2.27e-03(+)	0.0242±2.03e-03(+)	0.0497±5.53e-03(+)	0.0360±5.01e-03(+)	0.0304±3.55e-03(+)	0.0227±6.70e-04
	(5, 10)	0.0549±1.18e-02(+)	0.0281±4.23e-03(-)	0.0400±3.18e-03(+)	0.0363±3.15e-03(+)	0.0496±5.64e-03(+)	0.0395±5.31e-03(+)	0.0343±3.79e-03(=)	0.0348±5.90e-04
	(10, 5)	0.1241±2.96e-02(+)	0.0704±8.84e-03(+)	0.0665±7.10e-03(+)	0.0635±7.65e-03(+)	0.0687±6.94e-03(+)	0.1285±1.16e-02(+)	0.1155±6.37e-03(+)	0.0544±2.65e-03
	(5, 5)	0.1609±3.40e-02(+)	0.0849±1.19e-02(+)	0.1086±8.80e-03(+)	0.0817±5.78e-03(=)	0.0840±9.32e-03(+)	0.1361±1.45e-02(+)	0.1246±1.13e-02(+)	0.0820±3.06e-03
DF2	(10, 10)	0.0383±7.69e-03(+)	0.0275±4.63e-03(+)	0.0291±3.28e-03(+)	0.0275±2.69e-03(+)	0.0455±4.93e-03(+)	0.0385±3.45e-03(+)	0.0338±5.20e-03(+)	0.0227±8.83e-04
	(5, 10)	0.0465±1.12e-02(+)	0.0320±5.41e-03(+)	0.0413±4.82e-03(+)	0.0365±3.89e-03(+)	0.0470±5.54e-03(+)	0.0402±4.12e-03(+)	0.0368±6.13e-03(+)	0.0318±1.32e-03

TABLE I: Mean and standard deviation of MIGD obtained by the comparison algorithms in four dynamic environments.

Fig. 4: IGD curves for all algorithms on 14 test functions on dynamic environments ( $n_{t}=10$ and $\tau_{t}=10$ ).

Fig. 5: Average running time of all algorithms on some test functions on dynamic environments ( $n_{t}=10$ and $\tau_{t}=10$ ).

Fig. 6: MIGD values generated by different noise schedules $\alpha$ for all test functions in four dynamic environments. For example, DDM/L( $\alpha$ )-1 and DDM/C( $\alpha$ )-1 denote the linear and cosine schedules on the dynamic environment ( $n_{t}=10$ and $\tau_{t}=10$ ).

Fig. 7: Average running time and MIGD value for six test functions at different time steps on dynamic environments ( $n_{t}=10$ and $\tau_{t}=5$ ).

Algs.	$\text{ENV}_{1}$	$\text{ENV}_{2}$	$\text{ENV}_{3}$	$\text{ENV}_{4}$	$\text{ENV}_{5}$
KT-DMOEA	2.29e-01±(2.11e-02)	2.14e-01±(9.86e-02)	1.83e-01±(9.29e-02)	1.79e-01±(6.59e-02)	2.14e-01±(5.41e-02)
IGP-DMOEA	2.02e-01±(9.95e-02)	2.20e-01±(5.50e-02)	1.81e-01±(7.00e-02)	2.03e-01±(4.13e-02)	1.93e-01±(5.41e-02)
DIP-DMOEA	2.05e-01±(4.32e-02)	2.02e-01±(3.76e-02)	1.98e-01±(2.47e-02)	1.85e-01±(2.07e-02)	2.01e-01±(2.54e-02)
KTM-DMOEA	1.99e-01±(4.22e-02)	2.00e-01±(3.25e-02)	1.88e-01±(3.31e-02)	2.02e-01±(3.46e-02)	2.03e-01±(2.76e-02)
PSGN	2.57e-01±(1.07e-01)	2.39e-01±(1.05e-01)	1.89e-01±(1.05e-01)	2.31e-01±(6.41e-02)	2.39e-01±(6.58e-02)
VARE	1.88e-01±(5.47e-02)	2.06e-01±(7.38e-02)	1.86e-01±(7.19e-02)	2.16e-01±(5.80e-02)	1.88e-01±(3.27e-02)

TABLE II: Mean and standard deviation of HV obtained by the comparison algorithm for five environmental changes in the ROA problem.

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Problem	objectives	Dynamics	Remarks
DF1	2	mixed convexity-concavity,location of optima	dynamic PF and PS
DF2	2	switch of position-related variable, location of optima	static convex PF, dynamic PS, severe diversity loss
DF3	2	mixed convexity-concavity,variable-linkage, location of optima	dynamic PF and PS
DF4	2	variable-linkage, PF range, bounds of PS	dynamic PF and PS
DF5	2	number of knee regions, local of optima	dynamic PF and PS
DF6	2	mixed convexity-concavity, multimodality, location of optima	dynamic PF and PS

Appendix

TABLE S-I : Dynamic characteristics of 14 test functions in the CEC2018 dynamic multi-objective optimization competition.

Prob.	( $n_{t},\tau_{t}$ )	KT-DMOEA	IGP-DMOEA	DIP-DMOEA	KTM-DMOEA	PSGN	VARE	DM-DMOEA	DD-DMOEA
DF1	(10, 10)	0.5854±1.19e-02(+)	0.6079±6.27e-03(+)	0.6038±4.04e-03(+)	0.6089±3.67e-03(+)	0.4666±5.34e-03(+)	0.5783±7.94e-03(+)	0.6053±6.33e-03(+)	0.6096±1.29e-03
	(5, 10)	0.5605±1.52e-02(+)	0.5988±7.57e-03(=)	0.5854±6.22e-03(+)	0.5956±6.00e-03(+)	0.4689±5.97e-03(+)	0.5767±8.36e-03(+)	0.5973±7.54e-03(=)	0.5971±1.46e-03
	(10, 5)	0.4706±2.47e-02(+)	0.5366±1.07e-02(+)	0.5390±1.20e-02(+)	0.5457±1.10e-02(+)	0.4359±7.94e-03(+)	0.4469±1.42e-02(+)	0.5454±1.04e-02(+)	0.5547±3.89e-03
	(5, 5)	0.4358±2.95e-02(+)	0.5126±1.46e-02(=)	0.4731±1.33e-02(+)	0.5149±8.83e-03(=)	0.4220±1.13e-02(+)	0.4418±1.52e-02(+)	0.5204±1.55e-02(=)	0.5103±3.14e-03
DF2	(10, 10)	0.8125±1.10e-02(+)	0.8291±8.00e-03(+)	0.8245±5.74e-03(+)	0.8312±4.56e-03(+)	0.6634±6.50e-03(+)	0.8182±3.99e-03(+)	0.8368±8.88e-03(=)	0.8357±1.14e-03
	(5, 10)	0.8016±1.64e-02(+)	0.8214±1.01e-02(=)	0.7992±8.82e-03(+)	0.8127±6.28e-03(+)	0.6638±6.10e-03(+)	0.8134±5.73e-03(+)	0.8222±1.10e-02(=)	0.8184±2.17e-03

Appendix

TABLE S-II : Mean and standard deviation of MHV obtained by the comparison algorithms in four dynamic environments.

Fig. S-1 : IGD curves for all algorithms on 14 test functions on dynamic environments ( $n_{t}=5$ and $\tau_{t}=10$ ).

Fig. S-2 : IGD curves for all algorithms on 14 test functions on dynamic environments ( $n_{t}=10$ and $\tau_{t}=5$ ).

Fig. S-3 : IGD curves for all algorithms on 14 test functions on dynamic environments ( $n_{t}=5$ and $\tau_{t}=5$ ).

Fig. S-4 : Average running time of all algorithms on 14 test functions on dynamic environments ( $n_{t}=10$ and $\tau_{t}=10$ ).

Fig. S-5 : Average running time of all algorithms on 14 test functions on dynamic environments ( $n_{t}=10$ and $\tau_{t}=5$ ).

Fig. S-6 : Average running time of all algorithms on 14 test functions on dynamic environments ( $n_{t}=5$ and $\tau_{t}=10$ ).

Fig. S-7 : Average running time of all algorithms on 14 test functions on dynamic environments ( $n_{t}=5$ and $\tau_{t}=5$ ).

Prob.	$(n_{t},\tau_{t})$	DD-DMOEA/ $\text{v}_{1}$	DD-DMOEA $/\text{v}_{2}$	DD-DMOEA $/\text{v}_{3}$	DD-DMOEA
DF1	(10, 10)	0.0251±2.92e-03(+)	0.0394±4.01e-03(+)	0.0312±2.17e-03(+)	0.0235±2.09e-03
	(5, 10)	0.0374±3.86e-03(+)	0.0472±5.75e-03(+)	0.0404±3.76e-03(+)	0.0356±3.91e-03
	(10, 5)	0.0571±5.50e-03(=)	0.1338±1.39e-02(+)	0.0910±4.30e-03(+)	0.0576±5.87e-03
	(5, 5)	0.0934±7.35e-03(+)	0.1372±1.30e-02(+)	0.1130±1.14e-02(+)	0.0872±9.67e-03
DF2	(10, 10)	0.0264±4.45e-03(+)	0.0388±6.20e-03(+)	0.0253±4.03e-03(+)	0.0252±4.58e-03
	(5, 10)	0.0346±4.99e-03(+)	0.0392±2.89e-03(+)	0.0298±2.91e-03(-)	0.0327±5.70e-03

Appendix

TABLE S-III : Mean and standard deviation of MIGD values of all test functions for different strategies on four dynamic environments.

Prob.	$(n_{t},\tau_{t})$	DD-DMOEA/ $\text{v}_{1}$	DD-DMOEA $/\text{v}_{2}$	DD-DMOEA $/\text{v}_{3}$	DD-DMOEA
DF1	(10, 10)	0.6057±5.12e-03(+)	0.5800±6.93e-03(+)	0.5945±3.93e-03(+)	0.6081±3.66e-03
	(5, 10)	0.5904±7.15e-03(+)	0.5722±9.67e-03(+)	0.5829±8.10e-03(+)	0.5936±8.81e-03
	(10, 5)	0.5501±8.02e-03(=)	0.4359±1.69e-02(+)	0.4934±5.33e-03(+)	0.5486±8.66e-03
	(5, 5)	0.4953±1.25e-02(+)	0.4285±1.52e-02(+)	0.4607±1.75e-02(+)	0.5040±1.43e-02
DF2	(10, 10)	0.8302±6.77e-03(+)	0.8067±1.08e-02(+)	0.8343±4.35e-03(=)	0.8316±7.90e-03
	(5, 10)	0.8128±1.02e-02(+)	0.8070±5.20e-03(+)	0.8233±5.19e-03(=)	0.8159±1.08e-02

Appendix

TABLE S-IV : Mean and standard deviation of MHV values of all test functions for different strategies on four dynamic environments.

Prob.	$(n_{t},\tau_{t})$	DDM/L( $\alpha$ )-1	DDM/ $K_{1}$	DDM/ $K_{2}$	DDM/ $K_{3}$	DDM/ $K_{4}$	DDM/ $K_{5}$	DD-DMOEA
DF1	(10, 10)	0.0257±2.33e-03(+)	0.0251±2.01e-03(+)	0.0245±2.65e-03(+)	0.0249±2.28e-03(+)	0.0241±2.02e-03(+)	0.0254±2.75e-03(+)	0.0235±2.09e-03
	(5, 10)	0.0395±4.38e-03(+)	0.0419±4.53e-03(+)	0.0362±3.26e-03(+)	0.0363±3.55e-03(+)	0.0362±2.91e-03(+)	0.0369±5.08e-03(+)	0.0356±3.91e-03
	(10, 5)	0.0662±4.51e-03(+)	0.0619±6.57e-03(+)	0.0576±4.24e-03(=)	0.0598±6.94e-03(+)	0.0580±5.04e-03(+)	0.0603±3.99e-03(+)	0.0576±5.87e-03
	(5, 5)	0.0872±7.04e-03(=)	0.0944±6.28e-03(+)	0.0868±7.26e-03(=)	0.0888±8.88e-03(+)	0.0846±7.99e-03(=)	0.0812±6.30e-03(-)	0.0872±9.67e-03
DF2	(10, 10)	0.0267±3.26e-03(+)	0.0263±3.41e-03(+)	0.0256±2.87e-03(+)	0.0252±3.09e-03(=)	0.0249±4.55e-03(=)	0.0259±4.54e-03(+)	0.0252±4.58e-03
	(5, 10)	0.0375±4.59e-03(+)	0.0355±4.04e-03(+)	0.0351±4.97e-03(+)	0.0341±4.30e-03(+)	0.0351±4.92e-03(+)	0.0360±4.87e-03(+)	0.0327±5.70e-03

Appendix

TABLE S-V : Mean and standard deviations of MIGD values obtained from all test functions in four dynamic environments by different time steps

K

and noise scheduling

\alpha

Prob.	$(n_{t},\tau_{t})$	DDM/L( $\alpha$ )-1	DDM/ $K_{1}$	DDM/ $K_{2}$	DDM/ $K_{3}$	DDM/ $K_{4}$	DDM/ $K_{5}$	DD-DMOEA
DF1	(10, 10)	0.6035±4.29e-03(+)	0.6055±3.76e-03(+)	0.6069±4.15e-03(+)	0.6057±3.84e-03(+)	0.6069±3.89e-03(+)	0.6044±4.67e-03(+)	0.6081±3.61e-03
	(5, 10)	0.5849±7.85e-03(+)	0.5817±8.11e-03(+)	0.5921±6.73e-03(+)	0.5926±6.42e-03(+)	0.5925±5.09e-03(+)	0.5919±8.78e-03(+)	0.5936±8.21e-03
	(10, 5)	0.5345±5.43e-03(+)	0.5413±1.09e-02(+)	0.5512±5.77e-03(=)	0.5462±9.73e-03(+)	0.5488±7.76e-03(=)	0.5449±6.02e-03(+)	0.5484±8.81e-03
	(5, 5)	0.5022±1.16e-02(+)	0.4921±9.55e-03(+)	0.5046±1.07e-02(=)	0.5046±1.17e-02(=)	0.5104±1.23e-02(=)	0.5160±1.00e-02(=)	0.5040±8.76e-03
DF2	(10, 10)	0.8292±6.01e-03(+)	0.8306±5.78e-03(+)	0.8317±5.83e-03(=)	0.8313±5.19e-03(+)	0.8309±7.81e-03(+)	0.8307±7.20e-03(+)	0.8316±7.79e-03
	(5, 10)	0.8080±9.36e-03(+)	0.8115±8.34e-03(+)	0.8109±9.33e-03(+)	0.8124±9.09e-03(+)	0.8118±9.85e-03(+)	0.8102±9.12e-03(+)	0.8159±1.40e-02

Appendix

TABLE S-VI : Mean and standard deviations of MHV values obtained from all test functions in four dynamic environments by different time steps

K

and noise scheduling

\alpha

Prob.	$(n_{t},\tau_{t})$	DDM/ $\psi_{1}$	DDM/ $\psi_{2}$	DDM/ $\psi_{3}$	DDM/ $\psi_{4}$	DDM/ $\psi_{5}$	DDM/ $\psi_{6}$	DD-DMOEA
DF1	(10, 10)	0.0267±4.13e-03(+)	0.0247±3.89e-03(+)	0.0227±4.13e-03(=)	0.0286±4.26e-03(+)	0.0241±3.11e-03(+)	0.0247±3.42e-03(+)	0.0230±2.36e-03
	(5, 10)	0.0364±4.53e-03(+)	0.0359±5.33e-03(+)	0.0321±4.24e-03(-)	0.0403±6.02e-03(+)	0.0389±4.87e-03(+)	0.0399±5.97e-03(+)	0.0349±3.34e-03
	(10, 5)	0.0786±1.26e-02(+)	0.0727±1.07e-02(+)	0.0644±9.43e-03(+)	0.0826±6.54e-03(+)	0.0669±7.74e-03(+)	0.0646±5.91e-03(+)	0.0566±6.05e-03
	(5, 5)	0.1047±1.21e-02(+)	0.1035±1.02e-02(+)	0.0913±1.15e-02(+)	0.1131±1.57e-02(+)	0.1041±8.69e-03(+)	0.1042±1.18e-02(+)	0.0842±6.18e-03
DF2	(10, 10)	0.0380±6.97e-03(+)	0.0340±6.48e-03(+)	0.0291±5.23e-03(+)	0.0359±4.59e-03(+)	0.0336±4.80e-03(+)	0.0333±3.44e-03(+)	0.0243±4.27e-03
	(5, 10)	0.0391±4.75e-03(+)	0.0385±5.81e-03(+)	0.0363±6.82e-03(+)	0.0452±6.32e-03(+)	0.0457±5.42e-03(+)	0.0470±5.96e-03(+)	0.0317±7.79e-03

Appendix

TABLE S-VII : Mean and standard deviations of MIGD values obtained from all test functions in four dynamic environments by the standard deviation

\psi

of different Gaussian distributions.

Prob.	$(n_{t},\tau_{t})$	DDM/ $\psi_{1}$	DDM/ $\psi_{2}$	DDM/ $\psi_{3}$	DDM/ $\psi_{4}$	DDM/ $\psi_{5}$	DDM/ $\psi_{5}$	DD-DMOEA
DF1	(10, 10)	0.5985±7.46e-03(+)	0.5999±5.74e-03(+)	0.6041±6.85e-03(+)	0.6021±6.81e-03(+)	0.6057±5.21e-03(+)	0.6047±5.41e-03(+)	0.6088±3.61e-03
	(5, 10)	0.5841±1.04e-02(+)	0.5848±8.61e-03(+)	0.5917±7.71e-03(+)	0.5889±7.37e-03(+)	0.5862±8.81e-03(+)	0.5864±1.02e-02(+)	0.5963±8.21e-03
	(10, 5)	0.5149±1.01e-02(+)	0.5271±1.65e-02(+)	0.5407±1.35e-02(+)	0.5332±1.84e-02(+)	0.5428±1.17e-02(+)	0.5436±7.73e-03(+)	0.5511±8.81e-03
	(5, 5)	0.4886±1.89e-02(+)	0.4857±1.36e-02(+)	0.5016±1.56e-02(+)	0.4940±1.46e-02(+)	0.4907±1.29e-02(+)	0.4945±1.67e-02(+)	0.5075±8.76e-03
DF2	(10, 10)	0.8263±7.41e-03(+)	0.8251±8.75e-03(+)	0.8301±8.09e-03(+)	0.8284±9.65e-03(+)	0.8273±8.74e-03(+)	0.8271±5.77e-03(+)	0.8329±7.79e-03
	(5, 10)	0.8094±1.15e-02(+)	0.8109±1.14e-02(+)	0.8145±1.11e-02(+)	0.8145±9.36e-03(+)	0.8065±1.01e-02(+)	0.8035±1.25e-02(+)	0.8178±1.40e-02

Appendix

TABLE S-VIII : Mean and standard deviations of MHV values obtained from all test functions in four dynamic environments by the standard deviation

\psi

of different Gaussian distributions.

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