The Multi-Objective Polynomial Optimization.

This article focuses on multi-objective optimization problems (MOPs) defined by polynomial functions, specifically addressing the characterization, computation, and existence of Pareto points (PPs) and weakly Pareto points (WPPs). It establishes that Pareto values correspond to boundary points of a convex set derived from the epigraph of the polynomial objectives and provides semidefinite programming (SDP) representations when the polynomials are sum-of-squares (SOS) convex. The paper develops tight Moment-SOS relaxation methods to solve linear scalarization problems (LSPs) and Chebyshev scalarization problems (CSPs), including techniques to detect proper weights ensuring boundedness and to verify whether given points are (weakly) Pareto points. Additionally, it presents approaches to detect unboundedness in polynomial optimization, which is crucial for identifying nonexistence of proper weights or (weakly) Pareto points, and discusses open questions related to computational certificates for such nonexistence.