In mathematical optimization, ordinal optimization is the maximization of functions taking values in a partially ordered set ("poset"). Ordinal optimization has applications in the theory of queuing networks.
Contents
Definitions
A partial order is a binary relation "≤" over a set P which is reflexive, antisymmetric, and transitive, i.e., for all a, b, and c in P, we have that:
In other words, a partial order is an antisymmetric preorder.
A set with a partial order is called a partially ordered set (also called a poset). The term ordered set is sometimes also used for posets, as long as it is clear from the context that no other kinds of orders are meant. In particular, totally ordered sets can also be referred to as "ordered sets", especially in areas where these structures are more common than posets.
For a, b distinct elements of a partially ordered set P, if a ≤ b or b ≤ a, then a and b are comparable. Otherwise they are incomparable. If every two elements of a poset are comparable, the poset is called a totally ordered set or chain (e.g. the natural numbers under order). A poset in which every two elements are incomparable is called an antichain.
Examples
Standard examples of posets arising in mathematics include:
Extrema
There are several notions of "greatest" and "least" element in a poset P, notably:
For example, consider the natural numbers, ordered by divisibility: 1 is a least element, as it divides all other elements, but this set does not have a greatest element nor does it have any maximal elements: any g divides 2g, so 2g is greater than g and g cannot be maximal. If instead we consider only the natural numbers that are greater than 1, then the resulting poset does not have a least element, but any prime number is a minimal element. In this poset, 60 is an upper bound (though not the least upper bound) of {2,3,5} and 2 is a lower bound of {4,6,8,12}.
Additional structure
In many such cases, the poset has additional structure: For example, the poset can be a lattice or a partially ordered algebraic structure.
Lattices
A poset (L, ≤) is a lattice if it satisfies the following two axioms.
The join and meet of a and b are denoted by
It follows by an induction argument that every non-empty finite subset of a lattice has a join (supremum) and a meet (infimum). With additional assumptions, further conclusions may be possible; see Completeness (order theory) for more discussion of this subject.
A bounded lattice has a greatest (or maximum) and least (or minimum) element, denoted 1 and 0 by convention (also called top and bottom). Any lattice can be converted into a bounded lattice by adding a greatest and least element, and every non-empty finite lattice is bounded, by taking the join (resp., meet) of all elements, denoted by
A poset is a bounded lattice if and only if every finite set of elements (including the empty set) has a join and a meet. Here, the join of an empty set of elements is defined to be the least element
and
hold. Taking B to be the empty set,
and
which is consistent with the fact that
Ordered algebraic structure
The poset can be a partially ordered algebraic structure.
In algebra, an ordered semigroup is a semigroup (S,•) together with a partial order ≤ that is compatible with the semigroup operation, meaning that x ≤ y implies z•x ≤ z•y and x•z ≤ y•z for all x, y, z in S. If S is a group and it is ordered as a semigroup, one obtains the notion of ordered group, and similarly if S is a monoid it may be called ordered monoid. Partially ordered vector spaces and vector lattices are important in optimization with multiple objectives.
Ordinal optimization in computer science and statistics
Problems of ordinal optimization arise in many disciplines. Computer scientists study selection algorithms, which are simpler than sorting algorithms.
Statistical decision theory studies "selection problems" that require the identification of a "best" subpopulation or of identifying a "near best" subpopulation.
Applications
Since the 1960s, the field of ordinal optimization has expanded in theory and in applications. In particular, antimatroids and the "max-plus algebra" have found application in network analysis and queuing theory, particularly in queuing networks and discrete-event systems.