  We consider the problem of constructing an approximation of the Pareto curve
associated with the multiobjective optimization problem $\min_{\mathbf{x} \in
\mathbf{S}}\{ (f_1(\mathbf{x}), f_2(\mathbf{x})) \}$, where $f_1$ and $f_2$ are
two conflicting polynomial criteria and $\mathbf{S} \subset \mathbb{R}^n$ is a
compact basic semialgebraic set. We provide a systematic numerical scheme to
approximate the Pareto curve. We start by reducing the initial problem into a
scalarized polynomial optimization problem (POP). Three scalarization methods
lead to consider different parametric POPs, namely (a) a weighted convex sum
approximation, (b) a weighted Chebyshev approximation, and (c) a parametric
sublevel set approximation. For each case, we have to solve a semidefinite
programming (SDP) hierarchy parametrized by the number of moments or
equivalently the degree of a polynomial sums of squares approximation of the
Pareto curve. When the degree of the polynomial approximation tends to
infinity, we provide guarantees of convergence to the Pareto curve in
$L^2$-norm for methods (a) and (b), and $L^1$-norm for method (c).
