Vector optimization

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:

$C\mathrm{-}\underset{x\in S}{\min }f(x)$

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:

• $\overline{x}\in S$ is a weakly efficient point (weak minimizer) if for every $x\in S$ one has $f(x)-f(\overline{x})\in ̸-\mathrm{int}C$.
• $\overline{x}\in S$ is an efficient point (minimizer) if for every $x\in S$ one has $f(x)-f(\overline{x})\in ̸-C\mathrm{\setminus }\{0\}$.
• $\overline{x}\in S$ is a properly efficient point (proper 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 proper minimizer is a minimizer. And every minimizer is a weak minimizer.

Modern solution concepts not only consists of minimality notions but also take into account infimum attainment.

Solution methods

• Benson's algorithm for linear vector optimization problems.

Relation to multi-objective optimization

Any multi-objective optimization problem can be written as

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

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.