Vector optimization

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.

Solution methods

• Benson's algorithm for linear vector optimization problems

Relation to multi-objective optimization

Any multi-objective optimization problem can be written as

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

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.