# Vector optimization

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{{confused|Multi-objective optimization}}
'''Vector optimization''' is a subarea of [mathematical optimization](/source/mathematical_optimization) where [optimization problems](/source/Optimization_problem) with a vector-valued [objective function](/source/objective_function)s are optimized with respect to a given [partial ordering](/source/partial_ordering) and subject to certain constraints.  A [multi-objective optimization](/source/multi-objective_optimization) problem is a special case of a vector optimization problem: The objective space is the finite dimensional [Euclidean space](/source/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:
:<math>C\operatorname{-}\min_{x \in S} f(x)</math>
where <math>f: X \to Z</math> for a partially ordered [vector space](/source/vector_space) <math>Z</math>. The partial ordering is induced by a cone <math>C \subseteq Z</math>. <math>X</math> is an arbitrary set and <math>S \subseteq X</math> is called the feasible set.

== Solution concepts ==
There are different minimality notions, among them:
* <math>\bar{x} \in S</math> is a ''weakly efficient point'' (weak minimizer) if for every <math>x \in S</math> one has <math>f(x) - f(\bar{x}) \not\in -\operatorname{int} C</math>.
* <math>\bar{x} \in S</math> is an ''efficient point'' (minimizer) if for every <math>x \in S</math> one has <math>f(x) - f(\bar{x}) \not\in -C \backslash \{0\}</math>.
* <math>\bar{x} \in S</math> is a ''properly efficient point'' (proper minimizer) if <math>\bar{x}</math> is a weakly efficient point with respect to a [closed](/source/closure_(mathematics)) [pointed convex cone](/source/convex_cone) <math>\tilde{C}</math> where <math>C \backslash \{0\} \subseteq \operatorname{int} \tilde{C}</math>.

Every proper minimizer is a minimizer.  And every minimizer is a weak minimizer.<ref name="scalar2vector">{{Cite journal | last1 = Ginchev | first1 = I. | last2 = Guerraggio | first2 = A. | last3 = Rocca | first3 = M. | title = From Scalar to Vector Optimization | doi = 10.1007/s10492-006-0002-1 | journal = Applications of Mathematics | volume = 51 | pages = 5–36 | year = 2006 | url = https://irinsubria.uninsubria.it/bitstream/11383/1500550/1/am51-5-GinI-GueA-RocM-06.pdf | hdl = 10338.dmlcz/134627 | s2cid = 121346159 | hdl-access = free }}</ref>

Modern solution concepts not only consists of minimality notions but also take into account [infimum](/source/infimum) attainment.<ref name="Lohne">{{cite book|title=Vector Optimization with Infimum and Supremum|author=Andreas Löhne|publisher=Springer|year=2011|isbn=9783642183508}}</ref>

== Solution methods ==
* [Benson's algorithm](/source/Benson's_algorithm) for ''linear'' vector optimization problems.<ref name="Lohne">{{cite book|title=Vector Optimization with Infimum and Supremum|author=Andreas Löhne|publisher=Springer|year=2011|isbn=9783642183508}}</ref>

== Relation to multi-objective optimization ==
Any multi-objective optimization problem can be written as
:<math>\mathbb{R}^d_+\operatorname{-}\min_{x \in M} f(x)</math>
where <math>f: X \to \mathbb{R}^d</math> and <math>\mathbb{R}^d_+</math> is the non-negative [orthant](/source/orthant) of <math>\mathbb{R}^d</math>.  Thus the minimizer of this vector optimization problem are the [Pareto efficient](/source/Pareto_efficient) points.

== References ==
{{Reflist}}

Category:Mathematical optimization

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Adapted from the Wikipedia article [Vector optimization](https://en.wikipedia.org/wiki/Vector_optimization) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Vector_optimization?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.