  The multi-objective optimization is to optimize several objective functions
over a common feasible set. Since the objectives usually do not share a common
optimizer, people often consider (weakly) Pareto points. This paper studies
multi-objective optimization problems that are given by polynomial functions.
First, we study the convex geometry for (weakly) Pareto values and give a
convex representation for them. Linear scalarization problems (LSPs) and
Chebyshev scalarization problems (CSPs) are typical approaches for getting
(weakly) Pareto points. For LSPs, we show how to use tight relaxations to solve
them, how to detect existence or nonexistence of proper weights. For CSPs, we
show how to solve them by moment relaxations. Moreover, we show how to check if
a given point is a (weakly) Pareto point or not and how to detect existence or
nonexistence of (weakly) Pareto points. We also study how to detect
unboundedness of polynomial optimization, which is used to detect nonexistence
of proper weights or (weakly) Pareto points.
