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:

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

where $f:X\to Z$ for a partially ordered vector space $Z$. The partial ordering is induced by a cone $C\subseteq Z$. $X$ is an arbitrary set and $S\subseteq X$ is called the feasible set.

Solution concepts

There are different minimality notions, among them:

• ${\bar {x}}\in S$ is a weakly efficient point (weak minimizer) if for every $x\in S$ one has $f(x)-f({\bar {x}})\not \in -\operatorname {int} C$.
• ${\bar {x}}\in S$ is an efficient point (minimizer) if for every $x\in S$ one has $f(x)-f({\bar {x}})\not \in -C\backslash \{0\}$.
• ${\bar {x}}\in S$ is a properly efficient point (proper 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 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

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

where $f:X\to \mathbb {R} ^{d}$ and $\mathbb {R} _{+}^{d}$ is the non-negative orthant of $\mathbb {R} ^{d}$. Thus the minimizer of this vector optimization problem are the Pareto efficient points.