I’m trying to understand how complementary slackness arises from the optimality conditions of a penalty/barrier method.
I have the following problem:
$$\min_x \{\frac{1}{2} x^T Q x + c^T x – t \sum\limits_{i=1}^n log(y_i) \}\,\,\, s.t. Ax+y=b $$
$Q \in \mathbb{S}^{nxn}_{++}\, c \in \mathbb{R}^n\, A \in \mathbb{R}^{m x n} \, b \in \mathbb{R}^{m} $
I know that the optimality conditions are that the gradient must be orthogonal to the constrained region, which means $-\nabla f \in R(A^T)$ or $\nabla f = A^T w$ for some $w \in \mathbb{R}^m$.
I want to somehow show that the optimality condition $-\nabla f \in R(A^T)$ gives rise to the condition, $w_i*y_i=0\,\, \forall i\,\,$.
I think it has something to do with complementary slackness, but I am not sure. I know intuitively that $w_i*y_i=0\,\, \forall i\,\,$ must be true at any FONC (stationary) point. But I do not understand how it comes from optimality conditions.
How does it come from optimality conditions?
I understand how to do it generally for the quadratic function: How do I find KKT Conditions for the Quadratic Function?