Constrained Dikin-Langevin diffusion for polyhedra

Abstract

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.