Efficient warm-start generation for nonconvex sets

Develop a polynomial-time algorithm that, given a nonconvex compact set X ⊂ R^n, constructs an initial distribution supported on X that is M-warm with respect to the uniform distribution π ∝ 1_X (i.e., sup_x ρ(x)/π(x) ≤ M for polynomially bounded M).

Background

The main results require a warm start to ensure rapid mixing and controlled rejection-sampling complexity for the In-and-Out algorithm. For convex bodies, several warm-start strategies are known, but analogous methods for general nonconvex bodies are lacking.

The authors explicitly highlight the absence of an efficient procedure to obtain a warm start in the nonconvex setting as a remaining challenge.