Vector optimization

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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 \in S} 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

Relation to multi-objective optimization

Any multi-objective optimization problem can be written as

\mathbb{R}^d_+\operatorname{-}\min_{x \in M} f(x)

where f: X \to \mathbb{R}^d and \mathbb{R}^d_+ is the non-negative orthant of \mathbb{R}^d. Thus the minimizer of this vector optimization problem are the Pareto efficient points.

References

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