 Bernstein linear programming hierarchy for polynomial minimization

For $f$ of degree $d$ and $n\ge \max (1,d)$, compute its degree-$n$ Bernstein basis coefficients and solve $\max \lambda$ subject to $\lambda \le {\beta }_{k,n}$, $0\le k\le n$. This is a linear program with $n+1$ inequalities. The coefficients form a convex-combination enclosure of the polynomial values, so $v_n\le \min f$. Degree elevation of Bernstein coefficients makes the bounds monotone. For every $ϵ>0$, the polynomial $f-(\min f-ϵ)$ is strictly positive, so positive Bernstein coefficients for a strictly positive polynomial gives an eventual feasible bound $\min f-ϵ$. Hence $v_n$ converges to the true minimum. Earlier indices $n<d$ can use a one-inequality program at any common coefficient-based lower bound, preserving the stated constraint count for every positive index.