# Pareto optimum

### From Glossary

See multiple objectives.

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Multiple objectives |

The problem has more than one objective
function. Since these do not lie in a totally ordered set, a
solution is often defined as an *efficient point* (sometimes
called a *Pareto optimum*): is feasible, and there does not
exist another feasible point, , such that and
for some , where indexes the objective
functions, and we assume we are maximizing. This reduces to the
usual definition of an optimum when there is only one objective.

There have been two principle approaches to solving a multiple objective mathematical program (MOMP):

*weighted sum*: Maximize , where- lexicographically: and for This results in for which is lexicographically greater than (or equal to) any feasible solution. (Note: the order is fixed in advance.)

Both methods yield an efficient point (if one exists). Under
certain assumptions, both methods can be used to generate the set
of all efficient points, called the *efficient frontier*.