Can Someone Help with my Symbolic Optimization in Mathematica

[gsenel]

Hi Everyone,

I need to maximize
\[Pi] = R*\[Alpha] (1/\[Alpha] (\[Beta]/R)^(1/(
1 - \[Beta])) - (1 - \[Gamma])*\[Beta]^(2/(
1 - \[Beta]))*\[Alpha]^(\[Beta]/(
1 - \[Beta]))) - (1/\[Alpha] (\[Beta]/R)^(1/(
1 - \[Beta])) - (1 - \[Gamma])*\[Beta]^(2/(
1 - \[Beta]))*\[Alpha]^(\[Beta]/(1 - \[Beta])))

with respect to R (R will be in terms of alpha, beta and gamma) under the constraint that beta, alpha and gamma will all be between 0 and 1. How can I solve this symbolic optimization with Mathematica?

Your help is greatly appreciated...

 

[gsenel]

By the way, the above problem can be directly pasted to the Mathematica notebook...

 

[oswaler]

Hello,

Thank you for reaching out for help with your symbolic optimization in Mathematica. This type of problem can be solved using Mathematica's built-in function, Maximize. Here is an example of how you can use this function to solve your optimization problem:

First, define your objective function:

Pi = R*α (1/α (β/R)^(1/(1 - β)) - (1 - γ)*β^(2/(1 - β))*α^(β/(1 - β))) - (1/α (β/R)^(1/(1 - β)) - (1 - γ)*β^(2/(1 - β))*α^(β/(1 - β)))

Next, define your constraints:

0 <= β <= 1
0 <= α <= 1
0 <= γ <= 1

Now, use the Maximize function to find the maximum value of Pi with respect to R:

Maximize[{Pi, 0 <= β <= 1 && 0 <= α <= 1 && 0 <= γ <= 1}, R]

This will give you the maximum value of Pi and the corresponding value of R that maximizes it. You can also use the NMaximize function if you want a numerical solution.

I hope this helps you solve your optimization problem. If you have any further questions, please don't hesitate to ask. Best of luck!