Addressing real-world optimization problems becomes particularly challenging when analytic objective functions or constraints are unavailable. While numerous studies have addressed the issue of unknown objectives, limited research has focused on scenarios where feasibility constraints are not given explicitly. Overlooking these constraints can lead to spurious solutions that are unrealistic in practice. To deal with such unknown constraints, we propose to perform optimization within the data manifold using diffusion models. To constrain the optimization process to the data manifold, we reformulate the original optimization problem as a sampling problem from the product of the Boltzmann distribution defined by the objective function and the data distribution learned by the diffusion model. To enhance sampling efficiency, we propose a two-stage framework that begins with a guided diffusion process for warm-up, followed by a Langevin dynamics stage for further correction. Theoretical analysis shows that the initial stage results in a distribution focused on feasible solutions, thereby providing a better initialization for the later stage. Comprehensive experiments on a synthetic dataset, six real-world black-box optimization datasets, and a multi-objective optimization dataset show that our method achieves better or comparable performance with previous state-of-the-art baselines.

翻译：解决现实世界中的优化问题，当解析目标函数或约束不可用时尤为困难。尽管已有大量研究关注未知目标函数问题，但针对可行性约束未明确给出的场景的探索仍有限。忽略这些约束可能导致不切实际的虚假解。为处理此类未知约束，我们提出利用扩散模型在数据流形内执行优化。为使优化过程约束于数据流形，我们将原始优化问题重新表述为从玻尔兹曼分布（由目标函数定义）与扩散模型学习的数据分布之乘积中采样的采样问题。为提升采样效率，我们提出两阶段框架：首先通过引导扩散过程进行预热，随后利用朗之万动力学阶段进行进一步修正。理论分析表明，初始阶段生成的分布集中于可行解，从而为后续阶段提供更优的初始化。基于合成数据集、六个真实黑箱优化数据集及一个多目标优化数据集的综合实验表明，我们的方法在性能上与以往最先进基线相当或更优。