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We consider the problem of constructing an approximation of the Pareto curveassociated with the multiobjective optimization problem $\min_{\mathbf{x} \in\mathbf{S}}\{ (f_1(\mathbf{x}), f_2(\mathbf{x})) \}$, where $f_1$ and $f_2$ aretwo conflicting polynomial criteria and $\mathbf{S} \subset \mathbb{R}^n$ is acompact basic semialgebraic set. We provide a systematic numerical scheme toapproximate the Pareto curve. We start by reducing the initial problem into ascalarized polynomial optimization problem (POP). Three scalarization methodslead to consider different parametric POPs, namely (a) a weighted convex sumapproximation, (b) a weighted Chebyshev approximation, and (c) a parametricsublevel set approximation. For each case, we have to solve a semidefiniteprogramming (SDP) hierarchy parametrized by the number of moments orequivalently the degree of a polynomial sums of squares approximation of thePareto curve. When the degree of the polynomial approximation tends toinfinity, 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).

Approximating Pareto Curves using Semidefinite Relaxations