A multi-objective optimization problem is a special case of a vector optimization problem: The objective space is the finite dimensional Euclidean space partially ordered by the component-wise "less than or equal to" ordering.
In mathematical terms, a vector optimization problem can be written as: where
for a partially ordered vector space
The partial ordering is induced by a cone
[1] Modern solution concepts not only consists of minimality notions but also take into account infimum attainment.
[2] Any multi-objective optimization problem can be written as where
Thus the minimizer of this vector optimization problem are the Pareto efficient points.