Consider the (non-linear) optimization problem ($P$)
$$max \quad3x_1 + 4x_2$$
$$s.t. \quad x_1^2 + x_2^2 \leq 25$$
$$ \quad x_1,x_2 \geq 0$$
Formulate the Lagrangian function $\varTheta(y)$ and solve it for fixed $y>0$
This is the function that I obtained. $\varTheta(y)=25y_1 + max\{(3-y_1x_1)x_1 + (4-y_1x_2)x_2; \quad x_1,x_2\geq 0\}$
However, Since the coefficents in the $max$ function still have $x_i$'s in them, I am not sure how to proceed. Could anyone please help me out?
Since the objective function is increasing in the domain you just have to check the boundary an the problem reduces to $$max_{x_1,x_2}\quad 3x_1+4x_2$$ $$s.t.\quad x_1^2+x_2^2=25$$ which has the Lagrangian $$\varTheta(x_1,x_2,\lambda)=3x_1+4x_2-\lambda(x_1^2+x_2^2-25)$$ By applying first order conditions $$\frac{\partial \varTheta}{\partial x_1}=3-2\lambda x_1=0$$ $$\frac{\partial \varTheta}{\partial x_2}=4-2\lambda x_2=0$$ $$\frac{\partial \varTheta}{\partial \lambda}=-x_1^2-x_2^2+25=0$$ which can be solved by $x_1=3$, $x_2=4$ and $\lambda=1/2$.