# Optimal

For a mathematical program in standard form, $LaTeX: \textstyle x^* \in X$ (the domain) is an optimal solution if it is a maximum (or a minimum):

1. $LaTeX: x^*$ is feasible;
2. $LaTeX: f(x^*) \ge f(x)$ for all feasible $LaTeX: x$ (maximum value).

Some authors refer to an optimal solution when they mean a local optimum; others mean a member of the optimality region (which are global optima). In either case, the optimal value is the objective value, evaluated at an optimal solution. A solution is nearly optimal if it is feasible, but the optimality condition (2) is replaced by

$LaTeX: f(x^*) \ge f(x) - t \mbox{ for all feasible } x,$

where $LaTeX: t > 0.$ ($LaTeX: t = 0$ corresponds to being optimal). Typically, $LaTeX: t$ is specified as a small fraction, such as a cut-off tolerance for an algorithm to terminate finitely.