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:
where for a partially ordered vector space. The partial ordering is induced by a cone. is an arbitrary set and is called the feasible set.

Solution concepts

There are different minimality notions, among them:
  • is a weakly efficient point if for every one has.
  • is an efficient point if for every one has.
  • is a properly efficient point if is a weakly efficient point with respect to a closed pointed convex cone where.
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

Relation to multi-objective optimization

Any multi-objective optimization problem can be written as
where and is the non-negative orthant of. Thus the minimizer of this vector optimization problem are the Pareto efficient points.