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: $Z$
. The partial ordering is induced by a cone $C\subseteq Z$
. $S\subseteq X$

== Solution concepts == There are different minimality notions, among them: $f(x)-f({\bar {x}})\not \in -\operatorname {int} C$
. $f(x)-f({\bar {x}})\not \in -C\backslash \{0\}$
. $C\backslash \{0\}\subseteq \operatorname {int} {\tilde {C}}$
. 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 $\mathbb {R} ^{d}$
. Thus the minimizer of this vector optimization problem are the Pareto efficient points.

== References ==

Source

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Category

Optimization - Mathematics