Approximating Pareto Curves using Semidefinite Relaxations
This provides a systematic approach for approximating Pareto curves in optimization problems, which is incremental but offers theoretical guarantees for specific scalarization methods.
The paper tackles the problem of approximating Pareto curves in multiobjective optimization with polynomial criteria by developing a numerical scheme using semidefinite relaxations, achieving convergence guarantees in L^2-norm for two methods and L^1-norm for another as the polynomial degree increases.
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).