Optimizing Multivariate Function with Lagrange Multiplier Method

[hawaiifiver]

Homework Statement

Find the stationary value of

$$ f(u,v,w) = \left( \frac{c}{u} \right)^m + \left( \frac{d}{v} \right)^m + \left( \frac{e}{w} \right)^m $$

Constraint: $$ u^2 + v^2 + w^2 = t^2 $$

Note: $$ u, v, w > 0 $$. $$ c,d, e, t > 0 $$. $$ m > 0 $$ and is a positive integer.

Homework Equations

I have found the auxiliary function:

$$ F = \left( \frac{c}{u} \right)^m + \left( \frac{d}{v} \right)^m + \left( \frac{e}{w} \right)^m - \lambda ( u^2 + v^2 + w^2 - t^2) $$

and

$$ F_u = \frac{- m c^m}{ u^{m+1}} - 2\lambda u = 0 $$

$$ F_v = \frac{- m d^m}{ v^{m+1}} - 2\lambda v = 0 $$

$$ F_w = \frac{- m e^m}{ w^{m+1}} - 2\lambda w = 0 $$

The Attempt at a Solution

Its after this I am having difficult. I don't understand what to do. I assume I have to rearrange the above equations to get $$ u^2, v^2, w^2 $$ and substitute into the constraint, but I can't see how to do that because I get expressions for $$ u^{m + 2} , v^{m+2} , w^{m+2} $$ . Help please.

The Attempt at a Solution

Homework Statement

Homework Equations

The Attempt at a Solution

Homework Statement

Homework Equations

The Attempt at a Solution

Homework Statement

Homework Equations

The Attempt at a Solution

Homework Statement

Homework Equations

The Attempt at a Solution

Homework Statement

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

Homework Statement

Find the stationary value of

$$ f(u,v,w) = \left( \frac{c}{u} \right)^m + \left( \frac{d}{v} \right)^m + \left( \frac{e}{w} \right)^m $$

Constraint: $$ u^2 + v^2 + w^2 = t^2 $$

Note: $$ u, v, w > 0 $$. $$ c,d, e, t > 0 $$. $$ m > 0 $$ and is a positive integer.

Homework Equations

I have found the auxiliary function:

$$ F = \left( \frac{c}{u} \right)^m + \left( \frac{d}{v} \right)^m + \left( \frac{e}{w} \right)^m - \lambda ( u^2 + v^2 + w^2 - t^2) $$

and

$$ F_u = \frac{- m c^m}{ u^{m+1}} - 2\lambda u = 0 $$

$$ F_v = \frac{- m d^m}{ v^{m+1}} - 2\lambda v = 0 $$

$$ F_w = \frac{- m e^m}{ w^{m+1}} - 2\lambda w = 0 $$

Okay, that's good. What I would do is switch all of those "$\lambda$" terms to the right:
$\frac{-m c^m}{u^{m+1}}= 2\lambda u$
$\frac{-m d^m}{v^{m+1}}= 2\lambda v$
$\frac{-m d^m}{w^{m+1}}= 2\lambda w$

(In fact, to find extrema of f with constraint g= constant, I tend to think of the Lagrange multiplier condition as $\nabla f= \lambda \nabla g$ rather than using the "auxiliary function" $f+ \lambda g$. It gives the same result, of course.)

Now, since a particular value of $\lambda$ is not part of the solution start by eliminating $\lambda$ by dividing one equation by another. Dividing the first equation by the second, for example, gives
$$\left(\frac{c}{d}\right)^m\left(\frac{v}{u}\right)^{m+1}= \frac{u}{v}$$
which gives
$$\left(\frac{v}{u}\right)^{m+2}= \left(\frac{d}{c}\right)^m$$
and similarly for the other equations.

The Attempt at a Solution

Its after this I am having difficult. I don't understand what to do. I assume I have to rearrange the above equations to get $$ u^2, v^2, w^2 $$ and substitute into the constraint, but I can't see how to do that because I get expressions for $$ u^{m + 2} , v^{m+2} , w^{m+2} $$ . Help please.

 

[Ray Vickson]

Homework Statement

Find the stationary value of

$$ f(u,v,w) = \left( \frac{c}{u} \right)^m + \left( \frac{d}{v} \right)^m + \left( \frac{e}{w} \right)^m $$

Constraint: $$ u^2 + v^2 + w^2 = t^2 $$

Note: $$ u, v, w > 0 $$. $$ c,d, e, t > 0 $$. $$ m > 0 $$ and is a positive integer.

Homework Equations

I have found the auxiliary function:

$$ F = \left( \frac{c}{u} \right)^m + \left( \frac{d}{v} \right)^m + \left( \frac{e}{w} \right)^m - \lambda ( u^2 + v^2 + w^2 - t^2) $$

and

$$ F_u = \frac{- m c^m}{ u^{m+1}} - 2\lambda u = 0 $$

$$ F_v = \frac{- m d^m}{ v^{m+1}} - 2\lambda v = 0 $$

$$ F_w = \frac{- m e^m}{ w^{m+1}} - 2\lambda w = 0 $$

The Attempt at a Solution

Its after this I am having difficult. I don't understand what to do. I assume I have to rearrange the above equations to get $$ u^2, v^2, w^2 $$ and substitute into the constraint, but I can't see how to do that because I get expressions for $$ u^{m + 2} , v^{m+2} , w^{m+2} $$ . Help please.

The Attempt at a Solution

You have expressions for $u^{m + 2} , v^{m+2} , w^{m+2}$ in terms of λ. You can take (m+2) roots, to get $u, v, w$. You can then substitute these expressions into the constraint, to get a single, simple equation for λ. Actually, it is easier to re-write everything in terms of μ = -λ; that will eliminate a lot of minus signs in the equations and solutions.

RGV

 

[hawaiifiver]

Thank you both. I'm understanding this problem now.