Optimization Problem: Minimizing an Objective Function with a Constraint

[malcolm60]

Help -- Optimization Problem

Hello people,
I am working on certain energy optimization problems in multiprocessor systems.
My objective function is:

E= U*x / (1-yC)^2

where U and C are constants and x and y are independent variables. I need to minimize this function under the constrain that

x*y = L.
where L is another constant.

i.e. I need the values of x and y which minimize E
Can anyone help me with this?

Thanks for your time

 

[Hurkyl]

Well, you could use the constraint to reduce your objective function to have a single variable, then you can simply apply calc I.

 

[malcolm60]

I am sorry I forgot to mention that x and y are discrete variables.

 

[Hurkyl]

I'd bet the discrete answer is near the continuous answer!

 

[Hurkyl]

And a quick graph of E gives a very strong suggestion of the behavior of your objective function!

 

[malcolm60]

Thanks Hurky,
Is there some way to check that the solution with continuous variables is optimal even for discrete variables, just the method would do great, i can look it up.

Also what is the strong suggestion you observe from the graph of E?

Thanks

 

[Hurkyl]

Have you tried graphing it yet?

 

[Hurkyl]

Edit: if C * L is small, then ignore this post, its observation will be irrelevant.


Or, here's another way to look at it... x and y are both supposed to be positive integers, right? What kind of things are U, C, and L? Also positive integers?

If you're trying to minimize a fraction, what do you want to do to its numerator? What do you want to do to its denominator? Can you think of any good way to do that, respecting the constraint?

 

[malcolm60]

Yes I have graphed it. For the values of the constants I used
U=0.006
C=0.00146

it seems to be a plane

 

[Hurkyl]

I meant after using the constraint to reduce your objective function to a function in just one variable. (I'd suggest x)

 

[malcolm60]

Yes all the constants are positive. the thing is the optimum value I get for y is a function of C only. This does not seem correct as it suggests that y is a constant too.

 

[Hurkyl]

Hrm, good call, it does seem much better to put things in terms of y, not x!


It suggests the optimal value for y is a constant (but you knew that).

It's clear from the form of the objective function that the value of U is irrelevant: its net effect is simply to multiply your function by a fixed constant. I was surprised at first to hear that L is irrelevant too, but again when you're thinking in terms of y, the L just winds up being another multiplicative constant.


Anyways, if you got y = 1/(3c), I agree, that is the only local minimum. You also need to check the values at the "boundary": E is really small when y is very large, but E is very large when y is near zero.


If your discrete set is sufficiently refined, then it should be obvious from the shape of the graph that the optimal y has to either be 1/(3c) (rounded one way or the other), or the maximum possible value of y. If you want a rigorous proof, you could show it by analyzing where the objective function is increasing or decreasing.

 

[malcolm60]

Thanks Hurkyl,
Will try that out