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Vector optimization is a subarea of mathematical optimization where optimization problems with 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.

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

$C-\underset{x\in S}{min}f\left(x\right)$

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

There are different minimality notions, among them:

• $\overline{x}\in S$ is a weakly efficient point (weak minimizer) if for every $x\in S$ one has $f\left(x\right)-f\left(\overline{x}\right)\in ̸-\mathrm{int}C$.
• $\overline{x}\in S$ is an efficient point (minimizer) if for every $x\in S$ one has $f\left(x\right)-f\left(\overline{x}\right)\in ̸-C\setminus \left\{0\right\}$.
• $\overline{x}\in S$ is a properly efficient point (proper minimizer) if $\overline{x}$ is a weakly efficient point with respect to a closed pointed convex cone $\tilde{C}$ where $C\setminus \left\{0\right\}\subseteq \mathrm{int}\tilde{C}$.

Every proper minimizer is a minimizer. And every minimizer is a weak minimizer.^[1]

Modern solution concepts not only consists of minimality notions but also take into account infimum attainment.^[2]

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

Any multi-objective optimization problem can be written as

${ℝ}_{+}^d-\underset{x\in M}{min}f\left(x\right)$

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