Semidefinite Approximations of Conical Hulls of Measured Sets

Abstract

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