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:
where for a partially ordered vector space . The partial ordering is induced by a cone . is an arbitrary set and is called the feasible set.
Solution concepts There are different minimality notions, among them: is a weakly efficient point (weak minimizer) if for every one has . is an efficient point (minimizer) if for every one has . is a properly efficient point (proper minimizer) if is a weakly efficient point with respect to a closed pointed convex cone where .
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.
Solution methods Benson's algorithm for linear vector optimization problems.
Relation to multi-objective optimization Any multi-objective optimization problem can be written as
where and is the non-negative orthant of . Thus the minimizer of this vector optimization problem are the Pareto efficient points.
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