Vector optimization explained

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} 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.^[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.^[3]

Relation to multi-objective optimization

Any multi-objective optimization problem can be written as

R ^{d} _{+ \operatorname{ - } min} _{x} f(x) where f: X \to 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.

Notes and References

1. Ginchev . I. . Guerraggio . A. . Rocca . M. . From Scalar to Vector Optimization . 10.1007/s10492-006-0002-1 . Applications of Mathematics . 51 . 5–36 . 2006 . 10338.dmlcz/134627 . 121346159 . free .
2. Book: Vector Optimization with Infimum and Supremum. Andreas Löhne. Springer. 2011. 9783642183508.
3. Book: Vector Optimization with Infimum and Supremum. Andreas Löhne. Springer. 2011. 9783642183508.