  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.
