On the exactness for polynomial optimization strengthened with Fritz John conditions

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.