Computing the global infimum $f^$ of a multivariate polynomial subject to some constraints is a central question since it appears in many areas of engineering science. For some particular applications, it is of first importance to obtain reliable results. A lot of techniques has emerged to deal with constraints defined by polynomial inequalities. In this thesis, we focus on the optimization problem of a $n$-variate polynomial subject to constraints defined by $n$-variate polynomial equations. Our goal is to obtain reliable and efficient tools, algorithms and implementations to solve polynomial optimization problems. To do that, our strategy is to reduce the optimization problem subject to constraints defining algebraic sets of arbitrary dimension to an equivalent optimization problem, subject to constraints defining algebraic sets whose dimension is well-controlled. The algebraic variety defined by these new constraints is the union of the critical locus of the objective polynomial and an algebraic set of dimension at most 1. This is done by means of geometric objects defined as critical loci of linear projections. Since the dimension is well-controlled, the existence of certificates for lower bounds on $f^$ can be proved on this new variety. This is done by means of sums of squares and it does not require that $f^$ is reached. Likewise, we use the properties of our geometric objects to design an exact algorithm computing $f^$. If it exists, a minimizer is also returned. If there are $s$ constraints and if all the polynomials have degree at most $D$, its complexity is essentially cubic in $(sD)^n$ and linear in the evaluation complexity of the input. Its implementation, available as a Maple library, reflects the theoretical complexity. It solves problems unreachable by previous exact algorithms.
