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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

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.

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