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.

## Contents

* 1 Problem formulation
* 2 Solution concepts
* 3 Solution methods
* 4 Relation to multi-objective optimization
* 5 References

## Problem formulation

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

[math]\displaystyle{ C\operatorname{-}\min_{x \in S} f(x) }[/math]

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

## Solution concepts

There are different minimality notions, among them:

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

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

[math]\displaystyle{ \mathbb{R}^d_+\operatorname{-}\min_{x \in M} f(x) }[/math]

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

## References

1. ↑ Ginchev, I.; Guerraggio, A.; Rocca, M. (2006). "From Scalar to Vector Optimization". Applications of Mathematics 51: 5–36. doi:10.1007/s10492-006-0002-1. https://irinsubria.uninsubria.it/bitstream/11383/1500550/1/am51-5-GinI-GueA-RocM-06.pdf.
2. ↑ 2.0 2.1 Andreas Löhne (2011). Vector Optimization with Infimum and Supremum. Springer. ISBN 9783642183508.

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