Does this have an exact solution? Its for an engineering problem from my work and im a bit rusty on my Calculus and Algebra knowledge.

[Engineer98]

I would like to know if is this possible to do. Maybe i need more conditions, maybe not. The letters as usual are constants And i want to find an exact solution.
Being the sums of the variables presented on the sheet the closest numbers.

 

[blamocur]

This looks like a constrained optimization problem. Your equalities are constraints, whereas your approximate equalities and the homogeneousness requirement need to be translated to some goal function which you will minimize subject to the constraints.

 

[Cubist]

I would like to know if is this possible to do. Maybe i need more conditions, maybe not. The letters as usual are constants

Your hand written notes can be summarised by...
[math]x_i A_i + y_i B_i = C_i[/math][math]x_i + y_i \approx c[/math][math]E = \sum_{i=1}^n x_i[/math][math]D = \sum_{i=1}^n y_i[/math]For i in the range 1..n

Being the sums of the variables presented on the sheet the closest numbers.

I don't understand this sentence

And i want to find an exact solution.

Does this imply that you require the "≈" above to become exact "="?

 

[Engineer98]

This looks like a constrained optimization problem. Your equalities are constraints, whereas your approximate equalities and the homogeneousness requirement need to be translated to some goal function which you will minimize subject to the constraints.

It is, indeed, and optimization problem, i am having trouble finding that "goal equation" and then minimizing it

 

[Engineer98]

Your hand written notes can be summarised by...
[math]x_i A_i + y_i B_i = C_i[/math][math]x_i + y_i \approx c[/math][math]E = \sum_{i=1}^n x_i[/math][math]D = \sum_{i=1}^n y_i[/math]For i in the range 1..n


I don't understand this sentence



Does this imply that you require the "≈" above to become exact "="?

I want the solution of [math]x_i + y_i[/math] for each value of i to be the closest to each other. I know its unlikely for c to be the same for every equation but I want the set of variables that gives me for every c numbers like [28,29,30,30,27,28...] from each of the sums. Its a pretty vague constrain i know...

 

[blamocur]

It is, indeed, and optimization problem, i am having trouble finding that "goal equation" and then minimizing it

If we ignore the homogeneousness requirement we can add up the squares of the expressions which you want to be close to zero.

For homogeneousness I'd consider the distance from the [imath](x_1, ..., x_n, y_1, ..., y_n)[/imath] vector to the "homogeniousness axis" [imath]x_1 = x_2 = ... = x_n = y_1 = ... = y_n[/imath]. Another criterion can be the variance of the x,y set treated as a random sample.

But the choice of the goal function is more of an engineering problem rather than a mathematical one.