  We utilize the same technique as in [arXiv:2205.04254 (2022)] to provide some
representations of polynomials non-negative on a basic semi-algebraic set,
defined by polynomial inequalities, under more general conditions. Based on
each representation, we obtain semidefinite programs which return a sequence of
values that finitely converges to the optimal value of a given polynomial
optimization problem under generic assumption. Consequently, we can compute
exactly the minimal value of any polynomial over a basic convex semi-algebraic
set which is defined by the inequalities of concave polynomials.
