Method of substitution | Wiskunde op Tilburg University

Introduction: A constrained extremum problem is given by

Optimize $z(x,y)$
Subject to $g(x,y)=k$
Where $x \in D_1$ , $y \in D_2$

Method:

Rewrite $g(x,y)=k$ as a function $y(x)$ .
Replace $y$ in $z(x,y)$ by $y(x)$ : $Z(x)=z(x,y(x))$ .
Optimize $Z(x)$ as a function of one variable. This gives extremum location $c$ .
Since $Z(c)=z(c,d)$ , with $d=y(c)$ , it holds that $z(c,d)$ is the extremum of $z(x,y)$ .

Remark: The substitution cannot be used if

$g(x,y)=k$ cannot be written as a function

$y(x)$ (or

$x(y)$ ).