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I'm trying to follow Andrew Ng's notes on Support Vector Machines and had the following question.

In his notes, Ng, transforms the following optimization problem [using the notion of geometric margin] of the SVM

SVM Optimization Problem #1

into the following equivalent problem [using the notion of functional margin]

SVM Optimization Problem #2

My question is this: how are the conditions the same? I understand how $\gamma = \frac{\hat{\gamma}}{\Vert w\Vert}$, but what is the proof of the equivalence of the "s.t." conditions?

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In the first formulation, we're trying to maximize geometric margin. The condition

$$y^{(i)}(w^Tx^{(i)} + b) \geq \gamma, \forall i = 1, \dots, m$$

can be re-written as

$$y^{(i)}((\frac{w}{\|w\|})^Tx^{(i)} + \frac{b}{\|w\|}) \geq \gamma, \forall i = 1, \dots, m$$

since we've imposed an additional constraint in the first formulation, i.e. $\|w\|=1$. Now, in the second formulation we replace $\gamma$ with $\hat{\gamma}$, and can thus write

$$y^{(i)}((\frac{w}{\|w\|})^Tx^{(i)} + \frac{b}{\|w\|}) \geq \frac{\hat{\gamma}}{\|w\|}, \forall i = 1, \dots, m$$

since $\frac{\hat{\gamma}}{\|w\|} = \gamma$. Ensuring that the second constraint from the first formulation is met in the second formulation, we set $\|w\| = 1$ and can thus write

$$y^{(i)}(w^Tx^{(i)} + b) \geq \hat{\gamma}, \forall i = 1, \dots, m$$

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