How does the optimization problem
$$\min_{x,\eta} \frac{1}{2} x^TGx+x^Tc+M\eta$$ $$ s.t. Ax+\eta-b \geq0$$ $$\eta\geq0$$
look in standard form? What would the KKT look like?
The problem is, that I've got a solver that needs the standard $$\min_{x} \frac{1}{2} x^TGx+x^Tc$$ $$ s.t. Ax-b \geq0$$ as input format. I tried to use the substitution $y=[x \quad \eta]^T$. Then the problem is $$\min_{y} \frac{1}{2} y^T\begin{bmatrix}G & 0\\0 & 0\end{bmatrix}y+y^T\begin{bmatrix}c\\M\end{bmatrix}$$ $$ s.t. [A\quad 1]y-b \geq0$$ $$\eta\geq0$$ but with this problem the 'new' G- matrix $\begin{bmatrix}G & 0\\0 & 0\end{bmatrix}$ is singular and the solver can't solve the problem any more.