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Vector optimization is an optimization problem of simultaneously optimizing multiple objective functions subject to constraints and a given ordering. Any multi-objective optimization problem is a vector optimization problem with the trivial ordering.

Canonical example

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

C - \min_{x \in M} f (x)

where $f : X \to Z$ for some vector spaces $X, Z$ , $M \subseteq X$ , $C \subseteq Z$ is an ordering cone in $Z$ , and $C - \min$ denotes minimizing with respect to the ordering cone.

The solution to this minimization problem is the smallest set $S$ such that for every $s \in S$ there exists a $x \in M$ where $f (x) = s$ and $S + C \supseteq {f (x) : x \in M}$ .

Solution types

$\bar{x}$ is a weakly efficient point (w-minimizer) if there exists a neighborhood $U$ around $\bar{x}$ such that for every $x \in U$ it follows that $f (x) - f (\bar{x}) \in̸ - \int C$ .
$\bar{x}$ is an efficient point (e-minimizer) if there exists a neighborhood $U$ around $\bar{x}$ such that for every $x \in U$ it follows that $f (x) - f (\bar{x}) \in̸ - (C ∖ {0})$ .
$\bar{x}$ is a properly efficient point (p-minimizer) if $\bar{x}$ is a weakly efficient point with respect to a closed pointed convex cone $\tilde{C}$ where $C ∖ {0} \subseteq \int \tilde{C}$ .

Every p-minimizer is an e-minimizer. And every e-minimizer is a w-minimizer.^[1]

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

ℝ_{+}^{d} - \min_{x \in M} f (x)

where $f : X \to ℝ^{d}$ and $ℝ_{+}^{d}$ is the positive orthant of $ℝ^{d}$ . Thus the solution set of this vector optimization problem is given by the Pareto efficient points.

References

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↑ Template:Cite doi
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[scalar2vector-1] Template:Cite doi

[Lohne-2] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

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+'''Vector optimization''' is an [[optimization]] problem of simultaneously optimizing multiple [[objective function]]s subject to constraints and a given ordering.  Any [[multi-objective optimization]] problem is a vector optimization problem with the trivial ordering.
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+== Canonical example ==
+In mathematical terms, the vector optimization problem can be written as:
+:<math>C\operatorname{-}\min_{x \in M} f(x)</math>
+where <math>f: X \to Z</math> for some [[vector space]]s <math>X,Z</math>, <math>M \subseteq X</math>, <math>C \subseteq Z</math> is an [[ordering cone]] in <math>Z</math>, and <math>C\operatorname{-}\min</math> denotes minimizing with respect to the ordering cone.
+The solution to this minimization problem is the ''smallest'' set <math>S</math> such that for every <math>s \in S</math> there exists a <math>x \in M</math> where <math>f(x) = s</math> and <math>S + C \supseteq \{f(x): x \in M\}</math>.
+== Solution types ==
+* <math>\bar{x}</math> is a ''weakly efficient point'' (w-minimizer) if there exists a [[neighborhood (mathematics)|neighborhood]] <math>U</math> around <math>\bar{x}</math> such that for every <math>x \in U</math> it follows that <math>f(x) - f(\bar{x}) \not\in -\operatorname{int} C</math>.
+* <math>\bar{x}</math> is an ''efficient point'' (e-minimizer) if there exists a neighborhood <math>U</math> around <math>\bar{x}</math> such that for every <math>x \in U</math> it follows that <math>f(x) - f(\bar{x}) \not\in -(C \backslash \{0\})</math>.
+* <math>\bar{x}</math> is a ''properly efficient point'' (p-minimizer) if <math>\bar{x}</math> is a weakly efficient point with respect to a [[closure (mathematics)|closed]] [[convex cone|pointed convex cone]] <math>\tilde{C}</math> where <math>C \backslash \{0\} \subseteq \operatorname{int} \tilde{C}</math>.
+Every p-minimizer is an e-minimizer.  And every e-minimizer is a w-minimizer.<ref name="scalar2vector">{{cite doi|10.1007/s10492-006-0002-1}}</ref>
+== Solution methods ==
+* [[Benson's algorithm]] for ''linear'' vector optimization problems<ref name="Lohne">{{cite book|title=Vector Optimization with Infimum and Supremum|author=Andreas Löhne|publisher=Springer|year=2011|isbn=9783642183508}}</ref>
+== Relation to multi-objective optimization ==
+Any multi-objective optimization problem can be written as
+:<math>\mathbb{R}^d_+\operatorname{-}\min_{x \in M} f(x)</math>
+where <math>f: X \to \mathbb{R}^d</math> and <math>\mathbb{R}^d_+</math> is the positive [[orthant]] of <math>\mathbb{R}^d</math>.  Thus the solution set of this vector optimization problem is given by the [[Pareto efficient]] points.
+== References ==
+{{Reflist}}
+[[Category:Mathematical optimization]]

Kolmogorov–Zurbenko filter: Difference between revisions

Revision as of 22:37, 8 January 2014

Contents

Canonical example

Solution types

Solution methods

Relation to multi-objective optimization

References

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Kolmogorov–Zurbenko filter: Difference between revisions

Revision as of 22:37, 8 January 2014

Canonical example

Solution types

Solution methods

Relation to multi-objective optimization

References

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