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:
 * $$C\operatorname{-}\min_{x \in S} f(x)$$

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.

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

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.