  This paper aims to find efficient solutions to a multi-objective optimization
problem (MP) with convex polynomial data. To this end, a hybrid method, which
allows us to transform problem (MP) into a scalar convex polynomial
optimization problem (P$_z$) and does not destroy the properties of convexity,
is considered. First, we show an existence result for efficient solutions to
problem (MP) under some mild assumption. Then, for problem (P$_z$), we
establish two kinds of representations of non-negativity of convex polynomials
over convex semi-algebraic sets, and propose two kinds of finite convergence
results of the Lasserre-type hierarchy of semidefinite programming relaxations
for problem (P$_z$) under suitable assumptions. Finally, we show that finding
efficient solutions to problem (MP) can be achieved successfully by solving
hierarchies of semidefinite programming relaxations and checking a flat
truncation condition.
