# Optimization: transformation of target function and conditions

• January 15th 2013, 08:02 AM
Hess
Optimization: transformation of target function and conditions
In a recent microeconomics lecture I was confronted with the following problem:
maxx1, x2, x3 2*sqrt(x1)
s.t.
2*sqrt(x2y)=64
y=2*sqrt(2x3)
x1+x2+x3=112

the professor reformulated the problem without explanation to
maxx1, x2, x3 4x1
s.t.
16x22y22=644
y2=8x3
x1+x2+x3=112
and then solved by plugging the conditions into the target function. How can such a transformation be determined without changing the problem at hand? what is the logic or the reason behind this?

thanks for any clue
• January 15th 2013, 08:16 AM
hollywood
Re: Optimization: transformation of target function and conditions
If $2\sqrt{x_1}$ is at a maximum, then so is $4x_1=(2\sqrt{x_1})^2$. The changes made to the conditions are similar - taking the 4th power of both sides and squaring both sides.

- Hollywood