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:

[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

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