{{confused|Multi-objective 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: :<math>C\operatorname{-}\min_{x \in S} f(x)</math> where <math>f: X \to Z</math> for a partially ordered 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 pointed 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 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 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 of <math>\mathbb{R}^d</math>. Thus the minimizer of this vector optimization problem are the Pareto efficient points.
== References == {{Reflist}}
Category:Mathematical optimization