Vector optimization

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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#edit

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.

Solution concepts#edit

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]

Solution methods#edit

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

Relation to multi-objective optimization#edit

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.

References#edit

^ 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.

Categories Mathematical optimization

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