In this paper we consider the class of polynomial optimization problems with inequality and equality constraints, in which every problem of the class is obtained by perturbations of the objective function, while the constraint functions are kept fixed. Under certain assumptions, we establish some stability properties (e.g., strong H\"older stability with explicitly determined exponents, semicontinuity, etc.) of the global solution map, the Karush-Kuhn-Tucker set-valued map, and of the optimal value function for all problems in the class. It is shown that for almost every problem in the class, there is a unique optimal solution for which the global quadratic growth condition and the strong second-order sufficient conditions hold. Further, under local perturbations to the objective function, the optimal solution and the optimal value function (resp., the Karush-Kuhn-Tucker set-valued map) vary smoothly (resp., continuously) and the active constraints are constant. As a nice consequence, for almost all polynomial optimization problems, we can find a natural sequence of computationally feasible semidefinite programs, whose solutions give rise to a sequence of points in $\mathbb{R}^n$ converging to the optimal solution of the original problem.
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