# Method of substitution

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:
1. Rewrite $g(x,y)=k$ as a function $y(x)$.
2. Replace $y$ in $z(x,y)$ by $y(x)$: $Z(x)=z(x,y(x))$.
3. Optimize $Z(x)$ as a function of one variable. This gives extremum location $c$.
4. 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 $x(y)$).