Optimization constraint ($g(x)=0,h(x)\le 0$)

An optimization constraint restricts the feasible points in constrained optimization, for example by an equality $g(x)=0$ or an inequality $h(x)\le 0$. Extrema are considered on the resulting feasible set rather than on the whole domain of the objective.

For a constraint written $g_i(x)\ge 0$, its violation is the nonnegative number $v_i(x)=\max \{-g_i(x),0\}$. It vanishes precisely when the constraint holds. An equality $h_j(x)=0$ has violation $|h_j(x)|$. A norm of all these violations measures total infeasibility; its numerical magnitude depends on the scaling of the constraint functions.