Multivariable function optimization inconsistency

In summary, the conversation discusses a function that is dependent on 4 variables and the process of finding its minima in a specific domain. The conversation also includes a mistake in calculation and how it was solved.
  • #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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  • #2
Hi,

I have diffculty following the steps; perhaps you can post them ?

And I don't see how ##r_1(r_2^*(q),q)=0## can come out: ##f## does not exist for ##r_1 = 0## ...

##\ ##
 
  • #3
I had made some trivial mistake in calculation. Solved it now. Thanks.
 
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  • #4
RickRazor said:
I had made some trivial mistake in calculation. Solved it now. Thanks.
I'm glad you solved it.

Did it have to do with having r1 in the denominator in one instance then r2 in the denominator in the second instance?
 

1. What is a multivariable function optimization inconsistency?

A multivariable function optimization inconsistency occurs when there is a discrepancy between the expected optimal solution and the actual optimal solution of a function with multiple variables.

2. How does a multivariable function optimization inconsistency affect the accuracy of results?

A multivariable function optimization inconsistency can significantly impact the accuracy of results, as it indicates that the function may not have a unique optimal solution or that the optimization method used is not effective for the given function.

3. What are some possible causes of multivariable function optimization inconsistency?

There are several possible causes of multivariable function optimization inconsistency, including incorrect input parameters, inappropriate optimization method, or poorly designed function with multiple local optima.

4. How can multivariable function optimization inconsistency be addressed?

To address multivariable function optimization inconsistency, it is important to carefully select appropriate input parameters and optimization methods, as well as thoroughly analyze the function to identify any potential issues such as multiple local optima.

5. What are some techniques that can be used to prevent multivariable function optimization inconsistency?

Some techniques that can help prevent multivariable function optimization inconsistency include sensitivity analysis to determine the impact of input parameters on the optimal solution, using multiple optimization methods to compare results, and carefully designing the function to avoid multiple local optima.

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