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.
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.)