# Step-by-step example of solving a quadratic program with linear inequality constraints

I'm doing an exercise work about Support Vector Machines which involves solving a quadratic program of the form

\begin{aligned} & \underset{\boldsymbol\alpha \in \mathbb{R}^N}{\text{minimize:}} & & \frac{1}{2}\boldsymbol\alpha^TQ\boldsymbol\alpha +\textbf{p}^T\boldsymbol\alpha \\ & \text{subject to:} & & \; A\boldsymbol\alpha \geq \textbf{c}, \end{aligned}

where $Q$ is a positive-semidefinite matrix, $A$ a matrix and $\textbf{p}, \textbf{c}, \boldsymbol\alpha$ are vectors.

I've been reading about quadratic programming and about the methods of solving one but I haven't found any concrete pencil-and-paper example of how to actually solve this problem with, let's say, using the conjugate gradient method. In many textbooks I get directed into using online softwares for solving this problem but this doesn't fulfill my appetite, because I want to understand what is going on under the hood.

Could I get a concrete example or get directed to one, where I can see or implement myself all the steps needed to solve a quadratic program? The algorithm could be interior-point method for example.

Thank you!

• Conjugate gradients cannot be used on a constrained problem like this. You should be looking at information on interior-point methods or the simplex method for quadratic programming. – Michael Grant Mar 31 '15 at 13:58