I need to state the KKT conditions for the following problem:
Minimise $x_1^2 + 2x_2^2$ subject to $(x_1-1)^2 + x_2^2 \le 1$ and $x_2 = 1$.
I have that these conditions are:
$f(x^*) \le 0$
$h(x^*) = 0$
$\lambda^* \ge 0$
$\lambda^*_if_i(x^*) = 0$ where $1 \le i \le m$
$\nabla_xf(x^*) + \sum^m_{i=1}\lambda^*_i\nabla_xf_i^*(x^*) + \sum^l_{j=1}\mu^*_j\nabla_xh_j(x^*) = 0$
I have the first three conditions for this problem:
$(x_1-1)^2 + x_2^2 \le 1$
$x_2 = 1$
$\lambda \ge 0$
But I don't know how to approach the last two conditions. Please can someone help me understand how to achieve the final conditions.