The Cone Generated by Positive Semidefinite Mean Polynomials

Abstract: We study the cone $\mathcal{M}<em>{n,2d}$ of nonnegative mean polynomials---real $n$-variate forms of degree $2d$ that can be expressed as weighted power means $M</em>{q,p}(Y,w)$ with $q&gt;p$. This cone simultaneously generalises the cone of sums of squares $Σ<em>{n,2d}$ and the cone of sums of nonnegative circuit polynomials $\mathcal{C}</em>{n,2d}$. We prove that every square of an arbitrary polynomial belongs to the mean polynomial preprime $T_{\mathrm{mean}}$, that $T_{\mathrm{mean}}$ is strongly generating, and consequently that every polynomial strictly positive on a compact semialgebraic set admits a representation with mean polynomial certificates. We exhibit the Robinson form $\hat{R}$ as a separating example that lies in $\mathcal{M}<em>{4,4}$ but outside $\mathrm{SOSONC}</em>{4,4}$. Finally, we outline a convergent hierarchy of lower bounds for polynomial optimization based on the mean polynomial cone and discuss tractable depth-truncated approximations via signomial programming.