 # 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): $LaTeX: x^*$ is feasible, and there does not exist another feasible point, $LaTeX: x$, such that $LaTeX: f(x) \ge f(x^*)$ and $LaTeX: f_i(x) > f_i(x^*)$ for some $LaTeX: i$, where $LaTeX: i$ 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):

1. weighted sum: Maximize $LaTeX: w^T f(x)$, where $LaTeX: w > 0.$
2. lexicographically: $LaTeX: F_1 = \max \{f_1(x)\}$ and for $LaTeX: i > 1,$ $LaTeX: F_i = \max \{f_i(x): f_k(x) = F_k \mbox{ for } k=1, \ldots, i-1\}.$ This results in $LaTeX: x^*$ for which $LaTeX: f(x^*)$ 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.