Explictly solving constrained optimization problem (or, find a general solution on the trends)
Hi everybody,
For my research I need to solve a constrained optimization probelm. However, it belongs to a theoretic model, so I am looking for an explicit solution.
There are two variables, n and d. n≥1, 0≤d<$\frac{1}{2n} $
There are also two parameters, t and F. F only appears in the constraint. We have t,F>0 and t>16F.
The profit function has the following form:
$prof(n,d)$=$4td(a_{0}-a_{1})+16ntd^{2}[\frac{1}{2}(2a_{0}-1)(a_{0}-a_{1})-\frac{1}{8}]$ (to be maximised with respect to d)
$constraint(n,d):$ $(\frac{t}{n}-2td+4tda_{0})(4d(a_{1}-a_{0})+\frac{1}{n})-F=0 $
$a_{0} $ and $a_{1} $ depend on n, and are respectively equal to:
$a_{0}=\frac{-1}{2\sqrt{3}}[ \frac{1}{1-(2+\sqrt{3}) ^{n} } - \frac{1}{1-(2-\sqrt{3}) ^{n} }]$
$a_{1}=\frac{-1}{2\sqrt{3}}[ \frac{2+\sqrt{3} }{1-(2+\sqrt{3}) ^{n} } - \frac{2-\sqrt{3} }{1-(2-\sqrt{3}) ^{n} }]$
I have tried to find n as a function of d by solving the constraint, but I was not able to do it.
So, I consider the question solved if anyone is able to find the explicit solution to the problem that respects the domains of all variables and parameters.
However, I will also consider the question solved if this other problem is solved:
There is another function of interest in this model, which is
$CS(n,d)=v-\frac{5t}{4n}+2ntd^{2} $ (do not bother about $v$, it assumed to always have a value such that it does not create problems).
I ran a simulation and i saw that $prof(n,d)$ and $CS(n,d)$ have a specular trend as $d$ increases. As such, it seems like maximising $prof(n,d)$ with respect to $d$ implies minimising $CS(n,d)$. I attach a graph below to better show what I mean.
https://imgur.com/a/6VGmIvx
So, if anyone is able to demonstrate that when $prof(n,d)$ increases $CS(n,d)$ decreases, I will also consider the question as solved. Note that even in this case the constraint $constraint(n,d)=0$ must be respected.
If you need more clarifications, are more than welcome to give them to you!
Ffetpino
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So you want to maximize prof(n,d) with respect to d only? Is n known? The question is a little unclear. What are the known and unknown variables?
Hi Philip, t and F are known variables. the value of "n" needs to be obtained by solving the constraint. The resolution of the constraint should output something in the form n(d,t,F) At that point, by substituting n(d,t,F) in the prof function, I could find the maximum with respect to "d" under the domain 0
Due to the dependence of the numbers a0 and a1 on n it is very unlikely that a closed form of the solution can be achieved.
Hi Dynking, that is my gut feeling too, even if from the graph the trend of n as a function of d is so smooth and simple it bothers me (it is the blue curve in the imgur link). That is why I hoped that i was missing some semplifications or stuff like that. Still, you can try and work on the alternative solution regarding the two functions trends!
You can analytically show that either d=0 or d=1/2n, and simplify your optimization problem for a single variable optimization problem without constraint. I believe that's all that could be done. Would that be helpful?
Hi Philip, saldy that is not the case. As you can see from the IMGU link, the function maximum is somewhere in the middle between 0 and 1/(2n). As such, I cannot make assumptions on the value of d if I want to find its optimal value.