For polynomial optimization problems the sum-of-squares theory and Lasserre relaxation hierarchy provides a theoretically handy way of getting the solution. There are also results saying that finite relaxation is enough for global convergence.
My question is related to the unconstrained optimization of polynomials of the following form $$ J(x) = \sum_{k=1}^N q^2_k(x) $$ where $q_i(x)$ are arbitrary polynomials (they do have structure in my particular case, but I hope that wont be needed to get an answer).
Since $J(x)$ is sum of squares, it is non-negative. Let its minimum value be $$ j^* = \min_{x}J(x) $$
I know that there are general polynomials $P(x)$ such that $P(x)-p^*$ is not sum-of-squares. The questions is whether the assumption that $P(x)$ is sum-of-squares helps this situation or not.
Is $J(x)-j^*$ is also a sum of squares polynomial? If not, do we have a counterexample?
Any pointers towards relevant literature is also appreciated, as searching "is sum-of-squares" is useless for such specific questions.