Constrained Optimisation

$So$ far what $we$ had was unconstrained optimisation. $In$ other words, $we$ wanted $to$ find a local

maximum $or$ minimum, $or$ a saddle point $of$ a multivariate function without any restrictions $or$

constraints. However, sometimes $we$ want $to$ find a maximum $or$ a minimum $off(x,y)$ subject

$to$ a constraint $g(x,y)=k,$ where $kis$ a constant. $To$ solve this problem, $we$ use the Lagrange

Multiplier technique. $To$ find maximum and minimum values $off(x,y)$ subject $to$ the constraint

$g(x,y)=$kgradg$0ong(x,y)=kx,y,$ and $$ such that

gradf$(x,y)=gradg(x,y)$ and

$g(x,y)=k$

$(b)$ Evaluate $fx,yf$ notation $is$ used here. $Itis$ not $an$ eigenvalue.

Here, $itis$ a nonzero multiplier $-we$ are not interested $in$ its value, but $it$ will facilitate the

finding $of$ the critical points.$w=f(x,y,z)$ subject $to$ constraint $g(x,y,z)=0.$

Then you need $to$ solve the following system:

gradf$(x,y,z)=gradg(x,y,z)$

$g(x,y,z)=0$

This system has four equations and $4$ unknowns. However, generally, the equations are not

linear. Sometimes, you may use some mathematical tricks $to$ find the solution $to$ the system $of$

equations.