  Given a polynomial $f$ and a semi-algebraic set $S$, we provide a symbolic
algorithm 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, every
polynomial optimization problem whose optimum value is finite has an equivalent
form 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 is
finite. Given $d$ as the upper bound on the degrees of $f$ and polynomials
defining $S$, we prove that our method requires $O(d^{O(n)})$ arithmetic
operations to produce polynomials of degrees at most $d^{O(n)}$ defining
$\overline{f(S)}$.
