Lagrange

**Dogod11** · November 12th 2009, 04:43 AM

Hi good morning:

As the following is a nonlinear programming problem, I think solve it through Lagrange multipliers

but I have some problems:

The Levins' fitness sets offer an interesting application of the notion of convex sets in $R^n$ . A genotype of a particular species can have different "fitness", $W_1$ and $W_2$ ,
in two different environments. Assume that all genotypes of the species correspond to the points of convex set $W_1^2 + W_2^2 \leq{10}, 3W_2 - W_1 \geq{0}$ , and $3W_1 - W_2 \geq{0}$ .

What is the genotype (the set point) that maximizes:
$<br /> W_1?, W_2, W_1 + W_2?$

(These are the fittest genotypes in the environment first, the environment, secondly, and a mixed environment, respectively).

SOLUTION:

Since this is a nonlinear problem, Taking Lagrange multipliers, we:

$f(w_1,w_2,\lambda)=w_1+w_2-\lambda(w_1^2+w_2^2-10)$

Do:

$\frac{{\partial f}}{{\partial W_1}} = \frac{{\partial f}}{{\partial W_2}}= \frac{{\partial f}}{{\partial \lambda}} = 0$
$<br /> \frac{{\partial f}}{{\partial W_1}} = 1 - 2.\lambda.W_1 = 0$ (1)

$\frac{{\partial f}}{{\partial W_2}} = 1 + 2\lambda.W_2 = 0$ (2)

$\frac{{\partial f}}{{\partial \lambda}} = -2W_1.\lambda - 2W_2.\lambda + 10 = 0$ (3)

From Equation (1)

$W_1 = \displaystyle\frac{1}{2.\lambda}$

From Equation (2)

$W_2 = - \displaystyle\frac{1}{2.\lambda}$

From Equation (3)

$\lambda = 5(W_1 + W_2)$

I understand that there is now clear that replacing each variable in equations (1), (2), and (3),
To put everything in terms of $\lambda$ .

Find $\lambda$ and make the Hessian matrix, so it would be nice '?. This not see it, I very much appreciate if you help me finish this exercise

a greeting and thank you very much

**HallsofIvy** · November 12th 2009, 04:59 AM

Since $\lambda$ is not part of the data of the problem nor is it a value you are trying to find, I prefer to eliminate it as soon as possible. And you can typically do that by dividing one equation by another.

Here your equations are [tex]2\lambda W_1= 1[tex] and $2\lambda W_2= -1$ . Dividing the first equation by the second gives $\frac{W_1}{W_2}= -1$ or $W_1= -W_2$ .

With that, $W_1^2+ W_2^2= 10$ becomes $2W_1^2= 10$ so $W_1= \pm\sqrt{5}$ and then $W_2= \mp\sqrt{5}$ .

**Dogod11** · November 12th 2009, 05:18 AM

Hello, then the solution is that the maximum is at point $(\sqrt{5},\sqrt{5})$ and better $2\sqrt{5}$ For this restriction.

But I'm not sure what to do with the other, I have an idea:

Restrictions $\dfrac{1}{3}w_2\leq w_1\leq 3w_2$ only use it to check whether the maximum found on the edge of the circle, not in those limits.

if not then necessarily the maximum is reached at the intersections of the circle with the boundary lines:

I thank you once again your help,

greetings

Dogod11

**Dogod11** · November 12th 2009, 05:21 AM

Or perhaps I should also apply to the other two Lagrange constraints?

Greetings and thanks

**Dogod11** · November 12th 2009, 08:05 AM

Please could you help me again or someone with this problem?

I need a lot,

Thanks

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