The KKT condition is given by introducing relaxation parameters to the objective (derivative equals 0) so that whenever any element of $x$ is above $0$ or below $u$, parameters can increase to allow the system of equations to satisfy but at the same time cause penalties when x is not optimal.

So we have the following linear system that needs to be solved. $$ M=\begin{bmatrix} Q & E^\mathrm{T} & -I & I \\ E & 0 & 0 & 0\\ I & 0 & 0 & 0\\ I & 0 & 0 & 0\\ \end{bmatrix} \begin{bmatrix} x \\ \lambda \\ \mu \\ h \\ \end{bmatrix}=\begin{bmatrix} -q\\ b\\ 0\\ u\\ \end{bmatrix} $$

The KKT conditions are then the above equation for stationary, and complementary slackness (last two rows). Your listed conditions for primal feasibility. And $\lambda\ge0,\ \mu\ge0,\ h\ge0$ for dual feasibility.

This is my first ever answer on here. Hope I don't make a mistake.

The KKT condition is given by introducing relaxation parameters to the objective (derivative equals 0) so that whenever any element of $x$ is above $0$ or below $u$, parameters can increase to allow the system of equations to satisfy but at the same time cause penalties when x is not optimal.

So we have the following linear system that needs to be solved. $$ M=\begin{bmatrix} Q & E^\mathrm{T} & -I & I \\ E & 0 & 0 & 0\\ \end{bmatrix} \begin{bmatrix} x \\ \lambda \\ \mu \\ h \\ \end{bmatrix}=\begin{bmatrix} -q\\ b\\ \end{bmatrix} $$

The KKT conditions are then the above equation for stationary. Your listed conditions for primal feasibility.

complementary slackness:

$$\mu\odot x=0$$

$$h\odot(x-u)=0$$

And $\lambda\ge0,\ \mu\ge0,\ h\ge0$ for dual feasibility.

This is my first ever answer on here. Hope I don't make a mistake.

The KKT condition is given by introducing relaxation parameters to the objective (derivative equals 0) so that whenever any element of $x$ is above $0$ or below $u$, parameters can increase to allow the system of equations to satisfy but at the same time cause penalties when x is not optimal.

So we have the following linear system that needs to be solved. $$ M=\begin{bmatrix} Q & E^\mathrm{T} & -I & I \\ E & 0 & 0 & 0\\ I & 0 & 0 & 0\\ I & 0 & 0 & 0\\ \end{bmatrix} \begin{bmatrix} x \\ \lambda \\ \mu \\ h \\ \end{bmatrix}=\begin{bmatrix} -q\\ b\\ 0\\ u\\ \end{bmatrix} $$

This is my first ever answer on here. Hope I don't make a mistake.

The KKT condition is given by introducing relaxation parameters to the objective (derivative equals 0) so that whenever any element of $x$ is above $0$ or below $u$, parameters can increase to allow the system of equations to satisfy but at the same time cause penalties when x is not optimal.

So we have the following linear system that needs to be solved. $$ M=\begin{bmatrix} Q & E^\mathrm{T} & -I & I \\ E & 0 & 0 & 0\\ I & 0 & 0 & 0\\ I & 0 & 0 & 0\\ \end{bmatrix} \begin{bmatrix} x \\ \lambda \\ \mu \\ h \\ \end{bmatrix}=\begin{bmatrix} -q\\ b\\ 0\\ u\\ \end{bmatrix} $$

The KKT conditions are then the above equation for stationary, and complementary slackness (last two rows). Your listed conditions for primal feasibility. And $\lambda\ge0,\ \mu\ge0,\ h\ge0$ for dual feasibility.

This is my first ever answer on here. Hope I don't make a mistake.