Semidefinite approximations of conical hulls of measured sets

TL;DR

This paper studies semidefinite approximations of conical hulls generated by measures, proving convergence and providing bounds for specific cones like traveling salesman polytopes and nonnegative polynomials.

Contribution

It establishes convergence results for spectrahedral approximations of dual cones and develops tools to bound their distance, especially for symmetric cones.

Findings

Convergence of spectrahedral sequences to dual cones is proven.

Tools for bounding approximation errors are developed and applied.

Upper bounds are computed for specific cones like traveling salesman and polynomial cones.

Abstract

Let $C$ be a proper convex cone generated by a compact set which supports a measure $\mu$. A construction due to A.Barvinok, E.Veomett and J.B. Lasserre produces, using $\mu$, a sequence $(P_k)_{k\in \mathbb{N}}$ of nested spectrahedral cones which contains the cone $C^*$ dual to $C$. We prove convergence results for such sequences of spectrahedra and provide tools for bounding the distance between $P_k$ and $C^*$. These tools are especially useful on cones with enough symmetries and allow us to determine bounds for several cones of interest. We compute such upper bounds for semidefinite approximations of cones over traveling salesman polytopes and for cones of nonnegative ternary sextics and quaternary quartics.