Vector optimization is a subarea of
mathematical optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
where
optimization problems with a vector-valued
objective function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost ...
s are optimized with respect to a given
partial ordering
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
and subject to certain constraints. A
multi-objective optimization
Multi-objective optimization (also known as multi-objective programming, vector optimization, multicriteria optimization, multiattribute optimization or Pareto optimization) is an area of multiple criteria decision making that is concerned with ...
problem is a special case of a vector optimization problem: The objective space is the finite dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
. 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'' (weak minimizer) if for every
one has
.
*
is an ''efficient point'' (minimizer) if for every
one has
.
*
is a ''properly efficient point'' (proper minimizer) if
is a weakly efficient point with respect to a
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
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
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
attainment.
Solution methods
*
Benson's algorithm
Benson's algorithm, named after Harold Benson, is a method for solving multi-objective linear programming problems and vector linear programs. This works by finding the "efficient extreme points in the outcome set". The primary concept in Bens ...
for ''linear'' vector optimization problems.
Relation to multi-objective optimization
Any multi-objective optimization problem can be written as
:
where
and
is the non-negative
orthant
In geometry, an orthant or hyperoctant is the analogue in ''n''-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions.
In general an orthant in ''n''-dimensions can be considered the intersection of ''n'' mutua ...
of
. Thus the minimizer of this vector optimization problem are the
Pareto efficient points.
References
{{Reflist
Mathematical optimization