I am trying to solve the problem below using basically Theorem 2 that is presented in these notes. It describes the usual method of solving the problem of static optimization when you have equality and inequality constraints. The basic idea is to deal with the binding inequality constraints as they were equality constraints.
The problem is
$f(x,y)=x^2 + y^2$ subject to $2x+y\le 2$, $x\ge 0$ and $ y\ge 0$. Note that $f$ is continuous and the set is compact. Therefore, there exists a maximal and a minimal in this set. Furthermore, it is easy to see the the maximal is achieved when $y=2$ e $x=0$.
If I write the Lagrangean, I get
$L(\mu_1,\mu_2,\mu_3,x,y)=x^2 + y^2 -\mu_1 (2x+y-2) -\mu_2 (-x) -\mu_3 (-y).$
The first order conditions are
$\partial L/\partial x=2x -2\mu_1 + \mu_2 =0$
$\partial L/\partial y=2y -\mu_1 + \mu_3 =0$
The additional conditions are:
$\mu_1 (2x+y-2)=0$
$\mu_2 x=0$
$\mu_3 y=0$
Consider the situation where the first two constraints are binding, where I get $x=0$ and $y=2$. In this case, if I built the bordered hessian I would have to check n-m (m is the number of binding constraints) = 2 - 2 =0 last determinants. So, how to I check the second order conditions to this problem? Does it make sense?
A similar situation happens when the two last constraints are binding, i.e., $x=y=0$ and is this case we have got a minimal point.
Am I missing any assumption?
