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.^[1]

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

Solution methods

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

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.

References

^ Ginchev, I.; Guerraggio, A.; Rocca, M. (2006). "From Scalar to Vector Optimization" (PDF). Applications of Mathematics. 51: 5–36. doi:10.1007/s10492-006-0002-1. hdl:10338.dmlcz/134627. S2CID 121346159.
^ ^a ^b Andreas Löhne (2011). Vector Optimization with Infimum and Supremum. Springer. ISBN 9783642183508.

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