# Minimum/maximum

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.