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

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

 

where   for a partially ordered vector space  . The partial ordering is induced by a cone  .   is an arbitrary set and   is called the feasible set.

Solution concepts

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There are different minimality notions, among them:

  •   is a weakly efficient point (weak minimizer) if for every   one has  .
  •   is an efficient point (minimizer) if for every   one has  .
  •   is a properly efficient point (proper minimizer) if   is a weakly efficient point with respect to a closed pointed convex cone   where  .

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

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Relation to multi-objective optimization

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Any multi-objective optimization problem can be written as

 

where   and   is the non-negative orthant of  . Thus the minimizer of this vector optimization problem are the Pareto efficient points.

References

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  1. ^ Ginchev, I.; Guerraggio, A.; Rocca, M. (2006). "From Scalar to Vector Optimization" (PDF). Applications of Mathematics. 51: 5–36. doi:10.1007/s10492-006-0002-1. hdl:10338.dmlcz/134627. S2CID 121346159.
  2. ^ a b Andreas Löhne (2011). Vector Optimization with Infimum and Supremum. Springer. ISBN 9783642183508.