Project

Constraint Minimization Problem - Pen & Paper Calculation

Problem statement:

`5-(x-2)^2-2(y-1)^2` subject to `x+4y-3=0`

Solution

`L=f-λg` , where `f=5-(x-2)^2-2(y-1)^2` and `g=x+4y-3`

Thus, I solve

`L=5-(x-2)^2-2(y-1)^2-λ(x+4y-3)`

`L_x=-2(x-2)-λ=0→Eq^n I`

`L_y=-4(y-1)-4λ=0→Eq^n II`

`L_λ=x+4y-3=0→Eq^n III`

which gives

`x=-λ/2+2` from `Eq^n I` and `y=-λ+1` from `Eq^n II`

Substituting x and y equations in `Eq^n III` :

`-λ/2+2+4(1-λ)=3`

thus,

`λ=2/3` and `x=5/3` `y=1/3`

ensures a minimum subject to `g = 0`

Function is minimized

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