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

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 (7)

• Figure 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).
• Figure 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}$).
• Figure 3: Illustration of dividing objection space by the knee point. The number of subspaces here is 3.
• Figure 4: IGD curves for all algorithms on 14 test functions on dynamic environments ($n_t=10$ and $\tau_t=10$).
• Figure 5: Average running time of all algorithms on some test functions on dynamic environments ($n_t=10$ and $\tau_t=10$).