Definition: A function value $z(c,d)$ at a feasible point $(c,d)$ is a minimum of the constrained extremum problem
$$\begin{array}{ll}
\mbox{minimize}&z(x,y)\\
\mbox{subject to}&g(x,y)=k,\\
\mbox{where} & x \in D_1, y \in D_2,\\
\end{array}
$$
if for each feasible point $(x,y)$ in the neighborhood of $(c,d)$,
\[
z(c,d) \leq z(x,y).
\]
The point $(c,d)$ is called a minimum location of the constrained extremum problem.
A function value $z(c,d)$ at a feasible point $(c,d)$ is a maximum of the constrained extremum problem
$$\begin{array}{ll}
\mbox{maximize}&z(x,y)\\
\mbox{subject to}&g(x,y)=k,\\
\mbox{where} & x \in D_1, y \in D_2,\\
\end{array}
$$
if for each feasible point $(x,y)$ in the neighborhood of $(c,d)$,
\[
z(c,d) \geq z(x,y).
\]
The point $(c,d)$ is called a maximum location of the constrained extremum problem.
