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.

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

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

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

There are different minimality notions, among them:

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

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

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