  We present precise bit and degree estimates for the optimal value of the
polynomial optimization problem $f^*:=\text{inf}_{x\in \mathscr{X}}~f(x)$,
where $\mathscr{X}$ is a semi-algebraic set satisfying some non-degeneracy
conditions. Our bounds depend on the degree, the bitsize of $f$, and the
polynomials defining $\mathscr{X}$, and are single exponential with respect to
the number of variables. They generalize the single exponential bounds from
Jeronimo, Perrucci, and Tsigaridas (SIAM Journal on Optimization,
23(1):241--255, 2013) for the minimum of a polynomial function on a compact
connected component of a basic closed semi-algebraic set.
  The tools that we use allow us to obtain specialized bounds and dedicated
algorithms for two large families of polynomial optimization problems in which
the optimum value might not be attained. The first family forms a dense set of
real polynomial functions with a fixed collection of Newton polytopes; we
provide the best approximation yet for the bifurcation set, which contains the
optimal value, and we deduce an effective method for computations. As for the
second family, we consider any unconstrained polynomial optimization problem;
we present more precise bounds, together with a better bit complexity estimate
of an algorithm to compute the optimal value.
