Kolmogorov–Zurbenko filter

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:

$C\mathrm{-}{\min }_{x\in M}f(x)$

where $f:X\to Z$ for some vector spaces $X,Z$, $M\subseteq X$, $C\subseteq Z$ is an ordering cone in $Z$, and $C\mathrm{-}\min$ 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

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

Every p-minimizer is an e-minimizer. And every e-minimizer is a w-minimizer.

Solution methods

• Benson's algorithm for linear vector optimization problems

Relation to multi-objective optimization

Any multi-objective optimization problem can be written as

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

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

References

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