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 Z. The partial ordering is induced by a cone \( {\displaystyle C\subseteq Z} \) . X is an arbitrary set and \( S\subseteq X \) is called the feasible set.
Solution concepts
There are different minimality notions, among them:
\( {\displaystyle {\bar {x}}\in S} \) is a weakly efficient point (weak minimizer) if for every \( x\in S \) 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 \( x\in S one \) 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 \( {\bar {x}} \) is a weakly efficient point with respect to a closed pointed convex cone\( {\tilde {C}} \) 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 R\( \mathbb {R} ^{d} \). Thus the minimizer of this vector optimization problem are the Pareto efficient points.
References
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.
Andreas Löhne (2011). Vector Optimization with Infimum and Supremum. Springer. ISBN 9783642183508.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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