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Interdependent optimization of two functions?

S G
Posted 7 years ago

Hi, I have a linear function with number of variables f(x1 x2, x3, x4 ...) which I want to minimize wrt the variables x1, x2 ...such that the sum of variables x1+x2+x3... also remains minimum. I would be thankful for any suggestion regarding this interdependent double optimization problem. thanks SG
POSTED BY: S G
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I think you're basically asking about the fundamental problem of linear algebra, from a geometric perspective, maybe with a few extra constraints. You might be interested in the following:

Brief discussion of linear solving in $\mathbb{R}^n$

If you are studying this question from a computational perspective, using Mathematica, you should also be interested in the inner workings of the RowReduce function.

Thanks,

Brad
POSTED BY: Brad Klee
POSTED BY: Valeriu Ungureanu
S G
S G
Posted 7 years ago

Thanks a lot, it seems to be a good idea !
POSTED BY: S G
POSTED BY: Frank Kampas
POSTED BY: Brad Klee
S G
S G
Posted 7 years ago

Thanks fo sharing the info on Lagrange Multiplier method, I tought about it earlier, but seems Mathematica can not do it directly. I will reconsider it. In the meantime, in my last post (after your response) I have given a simple example of what I want to do. Please look at it and it might make more sense than what I posted initially. Thanks.
POSTED BY: S G
S G
S G
Posted 7 years ago

Here is the example again.. Let us consider a simple example z = Abs[4x + 6y - 18], where x, y >= 0. The minimum possible value for this function is 0, which can be achieved by a number of combinations of x and y. Among these possible sets (x, y), I want to find one for which x+y is minimum. In simple example as above, we can easily find that the solution to this problem is x = 0, y = 3. Any other set of x, y which satisfies above equation, x+y would always be higher than 3. Though I am not sure for a complicated problem having about 30 variables, how can I solve such problem using mathematica. thanks - SG
POSTED BY: S G
Bill Simpson
Bill Simpson
Posted 7 years ago

Since x,y>=0 just minimize the sum of your two goals

NMinimize[{Abs[4 x + 6 y - 18] + x + y, x >= 0, y >= 0}, {x, y}]

In your actual problem you might need Norm[]
POSTED BY: Bill Simpson
S G
S G
Posted 7 years ago

This seems to be a good idea, I will surely try. Though it would be great to think of a universal solution to such problems. Thanks a lot !!
POSTED BY: S G
S G
S G
Posted 7 years ago

Thanks for your response. Let us consider a simple example z = Abs[4x + 6y - 18], where x, y >= 0. The minimum possible value for this function is 0, which can be achieved by a number of combinations of x and y. Among these possible sets (x, y), I want to find one for which x+y is minimum. In simple example as above, we can easily find that the solution to this problem is x = 0, y = 3. Any other set of x, y which satisfies above equation, x+y would always be higher than 3. Though I am not sure for a complicated problem having about 30 variables, how can I solve such problem using mathematica. thanks - SG
POSTED BY: S G

I suggest you post a simple example of what you want to do and what you have done so far.
POSTED BY: Frank Kampas
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