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Given a polynomial $f$ and a semi-algebraic set $S$, we provide a symbolicalgorithm to find the equations and inequalities defining a semi-algebraic set$Q$ which is identical to the closure of the image of $S$ under $f$, i.e.,\begin{equation} Q=\overline{f(S)}\,. \end{equation} Consequently, everypolynomial optimization problem whose optimum value is finite has an equivalentform with attained optimum value, i.e., \begin{equation} \min \limits_{t\in Q}t =\inf\limits_{x\in S} f(x) \end{equation} whenever the right-hand side isfinite. Given $d$ as the upper bound on the degrees of $f$ and polynomialsdefining $S$, we prove that our method requires $O(d^{O(n)})$ arithmeticoperations to produce polynomials of degrees at most $d^{O(n)}$ defining$\overline{f(S)}$.

Semi-algebraic description of the closure of the image of a semi-algebraic set under a polynomial