Assume that we have two lines. Green and blue line.
$$w^Tx - b = 1$$ $$w^Tx - b = -1$$
Our goal is to find $w$ and $b$ here.
The first we can do, is to find the distance between these two lines.
We first find the ratio between $x^+$ and $x^−$. They are coordinates of $x$ that can describe
$$w^Tx^+ - b = 1$$ $$w^Tx^− - b = -1$$
So $x \in x^+, x^−$
The ratio between $x^+$ and $x^−$ is
$$x^+ = x^− + \lambda w$$
Where $\lambda$ is the distance between the two lines. If we using
$$w^Tx^+ - b = 1$$
and add in
$$x^+ = x^− + \lambda w$$
We will get this:
$$w^T(x^− + \lambda w) - b = 1$$ $$w^Tx^− + \lambda w^Tw - b = 1$$
And notice that $w^Tx^− -b$ is $-1$
$$-1 +\lambda w^Tw = 1$$
And now it will become
$$\lambda w^Tw = 2$$
We want to have the distance by it self
$$\lambda = \frac{2}{w^Tw}$$
And we need to make sure that $\lambda$ is always a positive number, so we are taking $$||\lambda|| = \frac{||2||}{||w||} = \frac{2}{||w||}$$
Question:
The number $||\lambda||$ need to be equal to 2, then the lines is optimized. Because the perdendicular distance between these two lines are 2.
The quadratic programming problem is:
$$\frac{1}{2}w^TQw + c^Tw$$
Where the constraints are:
$$Aw \leq d$$ $$w \geq 0$$
How can I describe the support vector machine formula, onto the quadratic programming objective function?