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

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

where ${\displaystyle f:X\to Z}$  for a partially ordered vector 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:

• ${\displaystyle {\bar {x}}\in S}$  is a weakly efficient point (weak minimizer) if for every ${\displaystyle x\in S}$  one has ${\displaystyle f(x)-f({\bar {x}})\not \in -\operatorname {int} C}$ .
• ${\displaystyle {\bar {x}}\in S}$  is an efficient point (minimizer) if for every ${\displaystyle x\in S}$  one has ${\displaystyle f(x)-f({\bar {x}})\not \in -C\backslash \{0\}}$ .
• ${\displaystyle {\bar {x}}\in S}$  is a properly efficient point (proper minimizer) if ${\displaystyle {\bar {x}}}$  is a weakly efficient point with respect to a closed pointed convex cone ${\displaystyle {\tilde {C}}}$  where ${\displaystyle C\backslash \{0\}\subseteq \operatorname {int} {\tilde {C}}}$ .

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

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

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