Vector optimization

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Vector optimization is an optimization problem of simultaneously optimizing multiple objective functions subject to constraints and a given ordering. Any multi-objective optimization problem is a vector optimization problem with the trivial ordering.

Canonical example

In mathematical terms, the vector optimization problem can be written as:

$C\operatorname {-}\min _{{x\in M}}f(x)$

where $f:X\to Z$ for some vector spaces $X,Z$ , $M\subseteq X$ , $C\subseteq Z$ is an ordering cone in $Z$ , and $C\operatorname {-}\min$ denotes minimizing with respect to the ordering cone.

The solution to this minimization problem is the smallest set $S$ such that for every $s\in S$ there exists a $x\in M$ where $f(x)=s$ and $S+C\supseteq \{f(x):x\in M\}$ .

Solution types

${\bar {x}}$ is a weakly efficient point (w-minimizer) if there exists a neighborhood $U$ around ${\bar {x}}$ such that for every $x\in U$ it follows that $f(x)-f({\bar {x}})\not \in -\operatorname {int}C$ .
${\bar {x}}$ is an efficient point (e-minimizer) if there exists a neighborhood $U$ around ${\bar {x}}$ such that for every $x\in U$ it follows that $f(x)-f({\bar {x}})\not \in -(C\backslash \{0\})$ .
${\bar {x}}$ is a properly efficient point (p-minimizer) if ${\bar {x}}$ is a weakly efficient point with respect to a closed pointed convex cone ${\tilde {C}}$ where $C\backslash \{0\}\subseteq \operatorname {int}{\tilde {C}}$ .

Every p-minimizer is an e-minimizer. And every e-minimizer is a w-minimizer.^[1]

Solution methods

Benson's algorithm for linear vector optimization problems^[2]

Relation to multi-objective optimization

Any multi-objective optimization problem can be written as

${\mathbb {R}}_{+}^{d}\operatorname {-}\min _{{x\in M}}f(x)$

where $f:X\to {\mathbb {R}}^{d}$ and ${\mathbb {R}}_{+}^{d}$ is the positive orthant of ${\mathbb {R}}^{d}$ . Thus the solution set of this vector optimization problem is given by the Pareto efficient points.

References

↑ Ginchev, I.; Guerraggio, A.; Rocca, M. (2006). "From Scalar to Vector Optimization". Applications of Mathematics 51: 5. doi:10.1007/s10492-006-0002-1.
↑ Andreas Löhne (2011). Vector Optimization with Infimum and Supremum. Springer. ISBN 9783642183508.

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