Vector optimization

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:
:$C\operatorname\min_ f(x)$
where $f: X \to Z$ for a partially ordered vector space $Z$. The partial ordering is induced by a cone $C \subseteq Z$. $X$ is an arbitrary set and $S \subseteq X$ is called the feasible set.

Solution concepts

There are different minimality notions, among them:
* $\bar \in S$ is a ''weakly efficient point'' (weak minimizer) if for every $x \in S$ one has $f(x) - f(\bar) \not\in -\operatorname C$.
* $\bar \in S$ is an ''efficient point'' (minimizer) if for every $x \in S$ one has $f(x) - f(\bar) \not\in -C \backslash \$.
* $\bar \in S$ is a ''properly efficient point'' (proper minimizer) if $\bar$ is a weakly efficient point with respect to a closed pointed convex cone $\tilde$ where $C \backslash \ \subseteq \operatorname \tilde$.

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
:$\mathbb^d_+\operatorname\min_ f(x)$
where $f: X \to \mathbb^d$ and $\mathbb^d_+$ is the non-negative orthant of $\mathbb^d$. Thus the minimizer of this vector optimization problem are the Pareto efficient points.

References

{{Reflist

Mathematical optimization