$\text{minimize } ||x-x_0||_2^2$ subject to $x \succeq_K 0$
Where $x_0 \in \Bbb R^n$ and $K$ is a proper cone.
If $z$ represents the positive Lagrange multiplier, then:
$x-x_0 = z$ is a KKT condition.
Can someone explain how $x-x_0 = z$ is derived?
I'm assuming it's from the KKT condition:
$$\nabla f_0(x^*) + \sum_{i=1}^mDf_i(x^*)^Tz^*_i + \sum_{i=1}^p v_i^* \nabla h_i(x^*) = 0$$
But I'm having trouble seeing how.
The first term from the optimization problem should be $2x - 2x_0$ and the second term should be $-I_n^Tz = -z$ so why isn't this $2x - 2x_0 - z = 0$?