- #1

RickRazor

- 17

- 3

- TL;DR Summary
- Missing conceptual detail in optimization problems

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I have a function dependent on 4 variables ##f(r_1,r_2,q_1,q)##. I'm looking to minimize this function in the domain ##0\leq r_1 \leq r_2 \leq 1## with respect to the variables ##r_1, r_2## and ##q_1##.

To find the minima, I first solved ##\frac{\partial f}{\partial r_1}=0## and ##\frac{\partial f}{\partial q_1}=0##, giving ##r_1^*(r_2,q)## and ##q_1^*(r_2,q)##. Now I have the function of the form ##f(r_1^*(r_2,q),r_2,q_1^*(r_2,q),q).##

Now I solved ##\frac{\partial f(r_1^*(r_2,q),r_2,q_1^*(r_2,q),q)}{\partial r_2}=0## for ##r_2^*(q)##.

So, the final function is of the form ##f(r_1^*(r_2^*(q),q),r_2^*(q),q_1^*(r_2^*(q),q),q)## which is fine. Now I see later that ##r_1^*(r_2^*(q),q)=0## and ##q_1^*(r_2^*(q),q)=0##.

So, if I directly look for the function ##f(0,r_2,0,q)## and it's minima wrt ##r_2##, it's giving a different result, i.e. I have

##\min_{r_2} f(0,r_2,0,q) \neq f(r_1^*(r_2^*(q),q),r_2^*(q),q_1^*(r_2^*(q),q),q)## even though ##r_1^*(r_2^*(q),q)=0## and ##q_1^*(r_2^*(q),q)=0##. Why is this the case? Are there other simple examples?

The function is

##f(r_1,r_2,q_1,q)=3r_1+r_2+q_1^2/r_1+2(q-q_1)^2/(r_2-r_1)## and

##f(0,r_2,0,q)=r_2+2(q-q_1)^2/r_2##

##r_2^*(q)=\sqrt{\frac{2}{3}}q##,

##r_1(r_2^*(q),q)=0## and ##q_1(r_2^*(q),q)=0##.

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