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 $\displaystyle R^n$. A genotype of a particular species can have different "fitness", $\displaystyle W_1$ and $\displaystyle W_2$,

in two different environments. Assume that all genotypes of the species correspond to the points of convex set$\displaystyle W_1^2 + W_2^2 \leq{10}, 3W_2 - W_1 \geq{0}$, and $\displaystyle 3W_1 - W_2 \geq{0}$.

What is the genotype (the set point) that maximizes:

$\displaystyle

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:

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

Do:

$\displaystyle \frac{{\partial f}}{{\partial W_1}} = \frac{{\partial f}}{{\partial W_2}}= \frac{{\partial f}}{{\partial \lambda}} = 0$

$\displaystyle

\frac{{\partial f}}{{\partial W_1}} = 1 - 2.\lambda.W_1 = 0$ (1)

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

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

From Equation (1)

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

From Equation (2)

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

From Equation (3)

$\displaystyle \lambda = 5(W_1 + W_2)$

I understand that there is now clear that replacing each variable in equations (1), (2), and (3),

To put everything in terms of $\displaystyle \lambda$.

Find $\displaystyle \lambda$ 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