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.

Vector optimization for rhino

Problem formulation

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

C - ⁡ min x ∈ S f ( x )

where f : X → Z for a partially ordered vector space Z . The partial ordering is induced by a cone C ⊆ Z . X is an arbitrary set and S ⊆ X is called the feasible set.

Solution concepts

There are different minimality notions, among them:

x ¯ ∈ S is a weakly efficient point (weak minimizer) if for every x ∈ S one has f ( x ) − f ( x ¯ ) ∉ − int ⁡ C .

x ¯ ∈ S is an efficient point (minimizer) if for every x ∈ S one has f ( x ) − f ( x ¯ ) ∉ − C ∖ { 0 } .

x ¯ ∈ S is a properly efficient point (proper minimizer) if x ¯ is a weakly efficient point with respect to a closed pointed convex cone C ~ where C ∖ { 0 } ⊆ int ⁡ C ~ .

Every proper minimizer is a minimizer. And every minimizer is a weak minimizer.

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

Solution methods

Benson's algorithm for linear vector optimization problems

Relation to multi-objective optimization

Any multi-objective optimization problem can be written as

R + d - ⁡ min x ∈ M f ( x )

where f : X → R d and R + d is the non-negative orthant of R d . Thus the minimizer of this vector optimization problem are the Pareto efficient points.

References

Vector optimization Wikipedia

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