We propose a reflection-free Langevin framework for sampling and optimization on compact polyhedra. The method is based on the inverse Hessian of the logarithmic barrier, which defines a Dikin--Langevin diffusion whose drift and noise adapt to the local interior-point geometry. We show that trajectories started in the interior remain feasible for all finite times almost surely, so the constrained domain is preserved without reflections or projections. For computation, we discretize the diffusion using the Euler--Maruyama scheme and apply a Metropolis--Hastings correction, yielding a sampler that targets the exact constrained distribution. We also propose an annealed interacting variant for nonconvex optimization. Numerically, the Metropolis-adjusted method outperforms both the Dikin random walk and standard MALA on anisotropic box-constrained Gaussians, and the interacting optimizer escapes suboptimal basins more reliably than the non-interacting method.

翻译：我们提出了一种无需反射的Langevin框架，用于紧致多面体上的采样与优化。该方法基于对数障碍函数的逆Hessian矩阵，定义了局域内点几何自适应调整漂移与噪声的Dikin-Langevin扩散。我们证明，从内部出发的轨迹几乎必然在任意有限时间内保持可行，从而无需反射或投影即可保证约束域的有效性。计算方面，采用Euler-Maruyama格式离散化该扩散，并施加Metropolis-Hastings修正，获得能够精确采样约束分布的目标采样器。同时，针对非凸优化问题，我们提出了一种退火交互变体。数值实验表明，在非各向同性有界高斯分布上，经Metropolis调整的方法性能优于Dikin随机游走及标准MALA；而交互优化器在逃离次优吸引子方面的可靠性显著优于非交互方法。