Vector optimization is a subarea of mathematical optimization where optimization problems with a vector-valued objective functions are optimized with respect to a given partial ordering and subject to certain constraints. 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. 
Problem formulation.
In mathematical terms, a vector optimization problem can be written as:
formula_1
where formula_2 for a partially ordered vector space formula_3. The partial ordering is induced by a cone formula_4. formula_5 is an arbitrary set and formula_6 is called the feasible set.
Solution concepts.
There are different minimality notions, among them:
Every proper minimizer is a minimizer. And every minimizer is a weak minimizer.
Modern solution concepts not only consists of minimality notions but also take into account infimum attainment.
Relation to multi-objective optimization.
Any multi-objective optimization problem can be written as
formula_17
where formula_18 and formula_19 is the non-negative orthant of formula_20. Thus the minimizer of this vector optimization problem are the Pareto efficient points.
References.
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