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:

${\displaystyle C\operatorname {-} \min _{x\in S}f(x)}$

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

Solution concepts

There are different minimality notions, among them:

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

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.[2]

Relation to multi-objective optimization

Any multi-objective optimization problem can be written as

${\displaystyle \mathbb {R} _{+}^{d}\operatorname {-} \min _{x\in M}f(x)}$

where ${\displaystyle f:X\to \mathbb {R} ^{d}}$ and ${\displaystyle \mathbb {R} _{+}^{d}}$ is the non-negative orthant of ${\displaystyle \mathbb {R} ^{d}}$. 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" (PDF). Applications of Mathematics. 51: 5. doi:10.1007/s10492-006-0002-1.
2. Andreas Löhne (2011). Vector Optimization with Infimum and Supremum. Springer. ISBN 9783642183508.