NLP Sampling: Combining MCMC and NLP Methods for Diverse Constrained Sampling

TL;DR

This paper tackles generating diverse solutions under hard nonlinear constraints by formulating NLP Sampling, which combines energy-based objectives with constraint indicators into a tractable target distribution $p(x)$. It introduces Restarting Two-Phase NLP Samplers that alternately seed candidate points, perform slack downhill steps, and execute interior sampling (via NHR and mRRT) within the feasible set, governed by the relaxed energy $F_{\\gamma\\mu}(x)=\\gamma f(x)+\\mu s(x)^T s(x)$. A new diversity metric, the Minimum Spanning Tree Score $MSTS_p$, enables quantitative assessment of mode coverage across analytic and robotic benchmarks, where Gauss-Newton downhill and interior samplers often outperform plain gradient methods, and restart seeding effects vary by problem domain. The work discusses conceptual limitations, such as the limited role of Lagrange parameters in NLP Sampling and the challenges of diffused NLP for data-free global transport, while outlining future directions that could blend data-free approaches with learning-based denoising to achieve global mode calibration. Collectively, the framework advances constrained sampling by bridging constrained optimization, MCMC, and robotics, with practical implications for robust, diverse planning under complex constraints.

Abstract

Generating diverse samples under hard constraints is a core challenge in many areas. With this work we aim to provide an integrative view and framework to combine methods from the fields of MCMC, constrained optimization, as well as robotics, and gain insights in their strengths from empirical evaluations. We propose NLP Sampling as a general problem formulation, propose a family of restarting two-phase methods as a framework to integrated methods from across the fields, and evaluate them on analytical and robotic manipulation planning problems. Complementary to this, we provide several conceptual discussions, e.g. on the role of Lagrange parameters, global sampling, and the idea of a Diffused NLP and a corresponding model-based denoising sampler.

Figures (9)

• Figure 1: Samples generated by different downhill and interior methods.
• Figure 2: NHR and Langevin for sampling from a box with interior energy $f=(x-(1,1))^2$, for varying $K_\text{burn}$. Captions report on Evaluations/Sample and Earth Mover Distance to ground truth. Plain MCMC is visually also very similar, but converges somewhat slower. (Evaluations per sample are roughly proportional to $K_\text{burn}$ as only a single sample $K_\text{sam}=1$ is taken after burn-in.)
• Figure 3: Performances on the modes problem. The top row shows generated samples in 2D (a-b) and 6D (c-d). The bottom row shows $\text{MSTS}_p$ with growing #samples.
• Figure 4: Samples generated by different downhill and interior methods for the Inverse Kinematics problem ($x \in \mathbb{R}^7$, only the first two dimensions displayed). Note the high evaluations per sample for Langevin methods in this highly non-linear setting.
• Figure 5: a) IK problem: reaching for the yellow target; b) similar, with a large cylindrical obstacle (implying non-linear collision inequalities and disconnected modes), c) a complex sequential manipulation planning problem over 4 consecutive configurations: start-of-box-push, end-of-box-push, reach-stick, and touch-target-with-stick -- overlaid in this solution illustration. See text for details.