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.{{Cite journal | last1 = Ginchev | first1 = I. | last2 = Guerraggio | first2 = A. | last3 = Rocca | first3 = M. | title = From Scalar to Vector Optimization | doi = 10.1007/s10492-006-0002-1 | journal = Applications of Mathematics | volume = 51 | pages = 5–36 | year = 2006 | url = https://irinsubria.uninsubria.it/bitstream/11383/1500550/1/am51-5-GinI-GueA-RocM-06.pdf | hdl = 10338.dmlcz/134627 | s2cid = 121346159 | hdl-access = free }}

Modern solution concepts not only consists of minimality notions but also take into account infimum attainment.{{cite book|title=Vector Optimization with Infimum and Supremum|author=Andreas Löhne|publisher=Springer|year=2011|isbn=9783642183508}}

Solution methods

Benson's algorithm for linear vector optimization problems.{{cite book|title=Vector Optimization with Infimum and Supremum|author=Andreas Löhne|publisher=Springer|year=2011|isbn=9783642183508}}

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

Category:Mathematical optimization