Question

Use KKT conditions to find the optimal solution to this problem: maximize x1 - x2, such that x1^2 + x2^2 <= 1

Answer 1

$x_1-x_2$ -> max

$x_1^2+x_2^2<=1$

Lagrange:

$F(x_1,x_2, \lambda)=x_1-x2-\lambda*(x_1^2+x_2^2-1^2)$

we obtain the stationary conditions

$1-2\lambda x_1=0\\1+2\lambda x_2=0$

subtracting the first from the second we have:

$2\lambda (x1+x2)=0=>x_1=-x_2$

substituting in the original equation we get

$x_1=0.707106781186548\\x_2=-0.707106781186548$

answer: $1.414213562373095$