Inner and Outer Approximations of Star-Convex Semialgebraic Sets

This letter considers the problem of approximating a semialgebraic set with a sublevel-set of a polynomial using sum-of-squares optimization. In this setting, it is standard to seek a minimum volume outer approximation or maximum volume inner approximation. This is made difficult by the lack of a known relationship between the coefficients of an arbitrary polynomial and the volume of its sublevel sets. Previous works have proposed heuristics based on the determinant and trace objectives commonly used in ellipsoidal fitting. We propose a novel objective which yields both an outer and an inner approximation while minimizing the ratio of their respective volumes. This objective is scale-invariant and easily interpreted. We provide justification for its use in approximating star-convex sets. Numerical examples demonstrate that the approximations obtained are often tighter than those returned by existing heuristics when applied to convex and star-convex sets. We also provide algorithms for establishing the star-convexity of a semialgebraic set by finding inner and outer approximations of its kernel.