Basic question on langrange multipliers

[Cyclops]

I am not sure how to visualize the following three equations

F1(P1)= 561 + 7.92 P1+ 0.00156 P1^2F2(P2)= 310 + 7.85 P2 + 0.00194 P2^2F3(P3) = 78 + 7.97 P3 + 0.00482P3^2

150 <= P1 <= 600
100 <= P2 <= 400
50 <= P3 <=200
P1+P2+P3 = 850

This is an optimisation problem with limits. Do I consider each of the functions on their own plane - for example F1(P1) would be on the xy plane. F2(P2) would be on the zx plane and F3(P3) would be on another plane. P1+P2+P3 would then be a plane that cuts each of them and we find the gradient vector from there.

 

[ehrenfest]

What exactly are you trying to optimize?

 

[Cyclops]

I am trying to minimise P1, P2,P3. F1&F2&F3 are cost functions. P1, P2 and P3 must be within the limits above and they must add up to 850. The question set is a lagrangrian multiplier question for generation despatch. What I question is, to visualize this problem - does each function have its own plane ie could we rewrite the question as y = 561 + 7.92 x1+ 0.00156 x1^2, z = 310 + 7.85 x2 + 0.00194 x2^2 and (some other plane)=78 + 7.97 x3 + 0.00482x3^2. The langrange is then the point on each plane where the plane
x1+x2+x3 = 850 cuts the other planes as long as these points are within the constraints given. I ask this because all the examples for lagrangian mulitpliers I have seen are in one plane - the xy plane

 

[EnumaElish]

I am not sure how to visualize the following three equations

F1(P1)= 561 + 7.92 P1+ 0.00156 P1^2


F2(P2)= 310 + 7.85 P2 + 0.00194 P2^2


F3(P3) = 78 + 7.97 P3 + 0.00482P3^2

150 <= P1 <= 600



100 <= P2 <= 400



50 <= P3 <=200



P1+P2+P3 = 850

This is an optimisation problem with limits. Do I consider each of the functions on their own plane - for example F1(P1) would be on the xy plane. F2(P2) would be on the zx plane and F3(P3) would be on another plane. P1+P2+P3 would then be a plane that cuts each of them and we find the gradient vector from there.

I am trying to minimise P1, P2,P3.

If you are trying to minimize the P's, why do you need the F's?

 

[Cyclops]

The F is just to state that it is function. F1(P1) means the function of P1

 

[EnumaElish]

Right, but what do these functions have anything to do with minimizing the P's?

Are you trying to minimize the F's instead?

If not, why are they part of the problem?

 

[Cyclops]

EnumaElish - it was just the way the assignment was written. The solution of the problem is straightforward - just plug in the values to the lagrange equation. My concern was how to imagine the solution space of the problem. I have found a book which in effect states that the solution space is all in one plane - ie the equations could be rewritten as
y1= 561 + 7.92 x1+ 0.00156 x1^2


y2= 310 + 7.85 x2 + 0.00194 x2^2


y3 = 78 + 7.97 x3 + 0.00482x3^2

I had not quite worked out what the solution space of the problem was - was it 2 dimensional or more dimensions. I do not know if that makes any sense. My apologies if I have wasted your time.

 

[EnumaElish]

The answer depends on what each of the F's represent.

For example I can think of the P's being input prices and each F being a cost function based on some implicit production technology. In this case I'd represent all three F's on the same axis.

Although unlikely, it is not impossible that each F represents a different dimension. For example, F1 = cost, F2 = utility, and F3 = profit. Even in this case it may be possible to represent all three quantities in terms of a common unit (e.g. dollars), and put them on the same axis.

 

[uart]

The answer depends upon a well defined question being posed. As yet none has been.

Cyclops, if you have some specific problem in mind, such as minimizing the sum of the F’s, or perhaps minimising the sum of the squares of the F’s or whatever then you have to specify that as part of the problem or otherwise it’s meaningless.