Vector optimization is a subarea ofmathematical optimization whereoptimization problems with a vector-valuedobjective functions are optimized with respect to a givenpartial ordering and subject to certain constraints. Amulti-objective optimization problem is a special case of a vector optimization problem: The objective space is the finite dimensionalEuclidean space partially ordered by the component-wise "less than or equal to" ordering.

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

${\displaystyle C\text{-}\underset{x\in S}{min}f(x)}$

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

There are different minimality notions, among them:

• ${\displaystyle \overline{x}\in S}$ is aweakly efficient point (weak minimizer) if for every${\displaystyle x\in S}$ one has${\displaystyle f(x)-f(\overline{x})\notin -\mathrm{int}C}$.
• ${\displaystyle \overline{x}\in S}$ is anefficient point (minimizer) if for every${\displaystyle x\in S}$ one has${\displaystyle f(x)-f(\overline{x})\notin -C\mathrm{\setminus }\{0\}}$.
• ${\displaystyle \overline{x}\in S}$ is aproperly efficient point (proper minimizer) if${\displaystyle \overline{x}}$ is a weakly efficient point with respect to aclosedpointed convex cone${\displaystyle \tilde{C}}$ where${\displaystyle C\mathrm{\setminus }\{0\}\subseteq \mathrm{int}\tilde{C}}$.

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 accountinfimum attainment.^[2]

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

Any multi-objective optimization problem can be written as

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

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

• Mathematical optimization